2473:
1711:
5057:
2468:{\displaystyle {\begin{alignedat}{3}6&=2^{1}(2^{2}-1)&&=1+2+3,\\28&=2^{2}(2^{3}-1)&&=1+2+3+4+5+6+7\\&&&=1^{3}+3^{3}\\496&=2^{4}(2^{5}-1)&&=1+2+3+\cdots +29+30+31\\&&&=1^{3}+3^{3}+5^{3}+7^{3}\\8128&=2^{6}(2^{7}-1)&&=1+2+3+\cdots +125+126+127\\&&&=1^{3}+3^{3}+5^{3}+7^{3}+9^{3}+11^{3}+13^{3}+15^{3}\\33550336&=2^{12}(2^{13}-1)&&=1+2+3+\cdots +8189+8190+8191\\&&&=1^{3}+3^{3}+5^{3}+\cdots +123^{3}+125^{3}+127^{3}\end{alignedat}}}
38:
7743:
3272:
10004:
2907:
964:
1367:= 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609 and 57885161 (sequence
461:(1194â1252) mentioned the next three perfect numbers (33,550,336; 8,589,869,056; and 137,438,691,328) and listed a few more which are now known to be incorrect. The first known European mention of the fifth perfect number is a manuscript written between 1456 and 1461 by an unknown mathematician. In 1588, the Italian mathematician
5562:, Chapter 16, he says of perfect numbers, "There is a method of producing them, neat and unfailing, which neither passes by any of the perfect numbers nor fails to differentiate any of those that are not such, which is carried out in the following way." He then goes on to explain a procedure which is equivalent to finding a
4729:
664:
3267:{\displaystyle {\begin{array}{rcl}6_{10}=&2^{2}+2^{1}&=110_{2}\\28_{10}=&2^{4}+2^{3}+2^{2}&=11100_{2}\\496_{10}=&2^{8}+2^{7}+2^{6}+2^{5}+2^{4}&=111110000_{2}\\8128_{10}=&\!\!2^{12}+2^{11}+2^{10}+2^{9}+2^{8}+2^{7}+2^{6}\!\!&=1111111000000_{2}\end{array}}}
5259:
is a natural number that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called
4462:
All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare. There are a number of results on perfect numbers that are actually quite easy to prove but nevertheless superficially impressive; some of them also come under
4556:
4074:
2632:
3881:
959:{\displaystyle {\begin{aligned}p=2&:\quad 2^{1}(2^{2}-1)=2\times 3=6\\p=3&:\quad 2^{2}(2^{3}-1)=4\times 7=28\\p=5&:\quad 2^{4}(2^{5}-1)=16\times 31=496\\p=7&:\quad 2^{6}(2^{7}-1)=64\times 127=8128.\end{aligned}}}
3319:
stated that Euclid's rule gives all perfect numbers, thus implying that no odd perfect number exists, but Euler himself stated: "Whether ... there are any odd perfect numbers is a most difficult question". More recently,
4453:... a prolonged meditation on the subject has satisfied me that the existence of any one such âits escape, so to say, from the complex web of conditions which hem it in on all sidesâwould be little short of a miracle.
3967:
3606:
61:, that is, divisors excluding the number itself. For instance, 6 has proper divisors 1, 2 and 3, and 1 + 2 + 3 = 6, so 6 is a perfect number. The next perfect number is 28, since 1 + 2 + 4 + 7 + 14 = 28.
445:
in his first-century book "On the creation" mentions perfect numbers, claiming that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect. Philo is followed by
4269:
669:
1716:
441:
itself has to be prime. He also says (wrongly) that the perfect numbers end in 6 or 8 alternately. (The first 5 perfect numbers end with digits 6, 8, 6, 8, 6; but the sixth also ends in 6.)
1231:(Alhazen) circa AD 1000 was unwilling to go that far, declaring instead (also without proof) that the formula yielded only every even perfect number. It was not until the 18th century that
4836:
1385:= 74207281, 77232917, and 82589933. Although it is still possible there may be others within this range, initial but exhaustive tests by GIMPS have revealed no other perfect numbers for
3766:
2692:
It follows that by adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the
4437:
7035:
465:
identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8.
4724:{\displaystyle {\frac {1}{6}}+{\frac {1}{3}}+{\frac {1}{2}}+{\frac {1}{1}}={\frac {1}{6}}+{\frac {2}{6}}+{\frac {3}{6}}+{\frac {6}{6}}={\frac {1+2+3+6}{6}}={\frac {2\cdot 6}{6}}=2}
2480:
1614:
1704:
1658:
457:(Book XI, Chapter 30) in the early 5th century AD, repeating the claim that God created the world in 6 days because 6 is the smallest perfect number. The Egyptian mathematician
8106:
4544:
4372:
3687:
2889:
651:
130:
4920:
2824:
2761:
1457:
1358:
1288:
1192:
551:
402:
3974:
5246:
5027:
87:, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors; in symbols,
6848:
4990:
3492:
3416:
3452:
453:, who adds the observation that there are only four perfect numbers that are less than 10,000. (Commentary on Genesis 1. 14â19). St Augustine defines perfect numbers in
157:
3332:, and it has been conjectured as well that there are no odd harmonic divisor numbers other than 1. Many of the properties proved about odd perfect numbers also apply to
337:
In about 300 BC Euclid showed that if 2 â 1 is prime then 2(2 â 1) is perfect. The first four perfect numbers were the only ones known to early
1496:
5436:
236:
5353:
4956:
1566:
1533:
1225:
1126:
1081:
1044:
999:
584:
435:
292:
1294:
between even perfect numbers and
Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the
312:
256:
10038:
5283:
3772:
7686:
5446:
1374:
8099:
3309:
1394:
489:
5791:
5745:
5036:
Every even perfect number ends in 6 or 28, base ten; and, with the only exception of 6, ends in 1 in base 9. Therefore, in particular the
3888:
8906:
8092:
5923:
1389:
below 109332539. As of
December 2018, 51 Mersenne primes are known, and therefore 51 even perfect numbers (the largest of which is
3522:
8901:
8916:
8896:
3316:
7732:
7538:
7382:
7209:
5721:
5694:
5647:
5558:
9609:
9189:
7929:
7742:
5113:
6244:
6147:
3336:, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist.
8911:
5767:
9695:
4209:
7679:
7573:
4867:. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form
9011:
9361:
8680:
8473:
2660:(after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with
9396:
9366:
9041:
9031:
7883:
6007:
KĂŒhnel, Ullrich (1950). "VerschĂ€rfung der notwendigen
Bedingungen fĂŒr die Existenz von ungeraden vollkommenen Zahlen".
9537:
8951:
8685:
8665:
7568:
5628:
5196:
4468:
9227:
7638:
9391:
10033:
9486:
9109:
8866:
8675:
8657:
8551:
8541:
8531:
7919:
7476:
6749:
6543:
5962:
5109:
9371:
5906:
10028:
9614:
9159:
8780:
8566:
8561:
8556:
8546:
8523:
7904:
7131:
The
Collected Mathematical Papers of James Joseph Sylvester p. 590, tr. from "Sur les nombres dits de Hamilton",
5192:
5078:
4737:
1617:
3700:
1295:
474:
323:
10048:
10043:
8599:
7672:
5796:
5590:
5056:
160:
8856:
7446:
Steuerwald, R. "VerschÀrfung einer notwendigen
Bedingung fĂŒr die Existenz einer ungeraden vollkommenen Zahl".
7377:. Chapman & Hall/CRC Pure and Applied Mathematics. Vol. 201. CRC Press. Problem 7.4.11, p. 428.
4851:
4377:
329:
It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.
5532:
9725:
9690:
9476:
9386:
9260:
9235:
9144:
9134:
8746:
8728:
8648:
8058:
7924:
7848:
7000:
6846:(1950). "Satze uber Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretisehe Probleme. II".
5303:
5288:
5100:
3329:
37:
1575:
9985:
9255:
9129:
8760:
8536:
8316:
8243:
7909:
7868:
6197:
Zelinsky, Joshua (July 2019). "Upper bounds on the second largest prime factor of an odd perfect number".
5576:
5298:
5087:
4494:
4446:
1663:
5637:
3499:
has at least 101 prime factors and at least 10 distinct prime factors. If 3 is not one of the factors of
3315:
It is unknown whether any odd perfect numbers exist, though various results have been obtained. In 1496,
9240:
9094:
9021:
8176:
7838:
7707:
4069:{\displaystyle {\frac {1}{q}}+{\frac {1}{p_{1}}}+{\frac {1}{p_{2}}}+\cdots +{\frac {1}{p_{k}}}<\ln 2}
1623:
31:
9949:
9589:
5184:. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a
4504:
4326:
3654:
2836:
598:
90:
7410:
5875:
5160:
gives various other kinds of numbers. Numbers where the sum is less than the number itself are called
4958:
with a power of two in a similar way to the construction of even perfect numbers from
Mersenne primes.
4870:
2774:
2711:
1407:
1308:
1238:
1142:
501:
352:
9882:
9776:
9740:
9481:
9204:
9184:
9001:
8670:
8458:
8430:
8012:
7914:
7405:
6396:
6259:
6162:
6084:
5375:
5096:
588:
167:
7618:
7586:
7563:
6043:
5786:
5227:
4999:
345:
noted 8128 as early as around AD 100. In modern language, Nicomachus states without proof that
9604:
9468:
9463:
9431:
9194:
9169:
9164:
9139:
9069:
9065:
8996:
8886:
8718:
8514:
8483:
8073:
8068:
7863:
7858:
7843:
7782:
7044:
6911:
5782:
5293:
5273:
5044:
442:
10003:
4968:
3460:
3384:
10007:
9761:
9756:
9670:
9644:
9542:
9521:
9293:
9174:
9124:
9046:
9016:
8956:
8723:
8703:
8634:
8347:
7997:
7992:
7953:
7873:
7853:
7495:
7434:
7320:
7285:
7250:
7242:
7114:
7088:
6881:
6717:
6588:
6412:
6386:
6224:
6206:
6024:
5633:
5358:
5256:
4860:
4148:
4144:
4128:
4112:
3641:
3423:
3325:
2696:) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because
454:
135:
8891:
6843:
1462:
9901:
9846:
9700:
9675:
9649:
9426:
9104:
9099:
9026:
9006:
8991:
8713:
8695:
8614:
8604:
8589:
8367:
8352:
8033:
7973:
7593:
7534:
7530:
7506:
Riele, H.J.J. "Perfect
Numbers and Aliquot Sequences" in H.W. Lenstra and R. Tijdeman (eds.):
7378:
7372:
7205:
7106:
7053:
6969:
6920:
6865:
6817:
6768:
6701:
5981:
5831:
5761:
5717:
5711:
5690:
5684:
5643:
5563:
5405:
5030:
4864:
3277:
2627:{\displaystyle T_{2^{p}-1}=1+{\frac {(2^{p}-2)\times (2^{p}+1)}{2}}=1+9\times T_{(2^{p}-2)/3}}
1499:
450:
338:
198:
7355:
7199:
5667:
5325:
4928:
1538:
1505:
1197:
1098:
1053:
1016:
971:
556:
407:
264:
9937:
9730:
9316:
9288:
9278:
9270:
9154:
9119:
9114:
9081:
8775:
8738:
8629:
8624:
8619:
8609:
8581:
8468:
8420:
8415:
8372:
8311:
8063:
8038:
7958:
7944:
7878:
7762:
7722:
7544:
7522:
7485:
7426:
7312:
7277:
7234:
7177:
7098:
6959:
6857:
6807:
6758:
6693:
6552:
6486:
6404:
6343:
6267:
6216:
6170:
6128:
6092:
6016:
5989:
5971:
5859:
5516:
5249:
5221:
5185:
5161:
5149:
5140:
3333:
3284:
1569:
54:
7065:
6981:
6932:
6877:
6829:
6780:
6713:
9913:
9802:
9735:
9661:
9584:
9558:
9376:
9089:
8946:
8881:
8851:
8841:
8836:
8502:
8410:
8357:
8201:
8141:
8048:
8043:
7968:
7962:
7899:
7797:
7787:
7717:
7548:
7061:
6977:
6928:
6873:
6825:
6776:
6709:
5993:
5910:
5181:
5177:
5165:
5069:
6684:
McDaniel, Wayne L. (1970). "The non-existence of odd perfect numbers of a certain form".
4486:
28 is also the only even perfect number that is a sum of two positive cubes of integers (
7079:
Yamada, Tomohiro (2019). "A new upper bound for odd perfect numbers of a special form".
6400:
6263:
6166:
6088:
6069:
4863:; that is, they cannot be represented as the difference of two positive non-consecutive
3328:
suggesting that indeed no odd perfect number should exist. All perfect numbers are also
9918:
9786:
9771:
9635:
9599:
9574:
9450:
9421:
9406:
9179:
8876:
8831:
8708:
8306:
8301:
8296:
8268:
8253:
8166:
8151:
8129:
8116:
8053:
8007:
7817:
7807:
7777:
5499:
5278:
5224:
associated with a perfect number is a constant sequence. All perfect numbers are also
5157:
4843:
The number of divisors of a perfect number (whether even or odd) must be even, because
4464:
3321:
1232:
1228:
1006:
1002:
462:
319:
315:
297:
241:
7664:
6964:
6947:
5370:, and both 23 and 89 yield a remainder of 1 when divided by 22. Furthermore, whenever
2912:
10022:
9841:
9825:
9766:
9720:
9416:
9401:
9311:
9036:
8594:
8463:
8425:
8382:
8263:
8248:
8238:
8196:
8186:
8161:
8002:
7802:
7792:
7772:
7523:
7438:
7351:
7324:
7289:
7254:
7118:
6885:
6721:
6028:
5855:
5663:
5512:
5060:
3876:{\displaystyle \alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq {\frac {99k-224}{37}}}
1010:
77:
46:
7630:
6763:
6741:
6557:
6538:
6348:
6228:
6119:
Konyagin, Sergei; Acquaah, Peter (2012). "On Prime
Factors of Odd Perfect Numbers".
5976:
9877:
9866:
9781:
9619:
9594:
9511:
9411:
9381:
9356:
9340:
9245:
9212:
8961:
8935:
8846:
8785:
8362:
8258:
8191:
8171:
8146:
8017:
7934:
7812:
7757:
7727:
7596:
6996:
6416:
5834:
5261:
5122:
5037:
4923:
2693:
1398:
1305:
distributed computing project has shown that the first 48 even perfect numbers are
658:
73:
7466:
Nankar, M.L.: "History of perfect numbers," Ganita
Bharati 1, no. 1â2 (1979), 7â8.
6408:
6295:
6272:
6175:
6097:
5500:"A proof that all even perfect numbers are a power of two times a Mersenne prime"
5176:. A pair of numbers which are the sum of each other's proper divisors are called
9836:
9711:
9516:
8980:
8871:
8826:
8821:
8571:
8478:
8377:
8206:
8181:
8156:
7646:
6903:
6574:"On inequalities involving counts of the prime factors of an odd perfect number"
6573:
6441:
6148:"The second largest prime divisor of an odd perfect number exceeds ten thousand"
259:
84:
69:
6328:
1381:
Three higher perfect numbers have also been discovered, namely those for which
17:
9973:
9954:
9250:
8861:
7650:
6997:"Some results concerning the non-existence of odd perfect numbers of the form
6861:
6812:
6795:
6491:
6474:
6371:
6245:"The third largest prime divisor of an odd perfect number exceeds one hundred"
6220:
6132:
5740:
5173:
1132:
458:
342:
65:
8084:
7430:
7303:
Kanold, HJ (1956). "Eine
Bemerkung šuber die Menge der vollkommenen zahlen".
7110:
7057:
6973:
6924:
6869:
6821:
6772:
6705:
5985:
1091:
is not a prime number. In fact, Mersenne primes are very rare: of the primes
9579:
9506:
9498:
9303:
9217:
8335:
7767:
7621:
A projected distributed computing project to search for odd perfect numbers.
7601:
6507:
5839:
1291:
322:
proved that all even perfect numbers are of this form. This is known as the
7225:
Jones, Chris; Lord, Nick (1999). "Characterising non-trapezoidal numbers".
7102:
6616:
5575:
Commentary on the Gospel of John 28.1.1â4, with further references in the
4175:
must be smaller than an effectively computable constant depending only on
9680:
7417:
Kanold, H.-J. (1941). "Untersuchungen ĂŒber ungerade vollkommene Zahlen".
5954:
5592:
The Reception of Philonic Arithmological Exegesis in Didymus the Blind's
595:
For example, the first four perfect numbers are generated by the formula
4501:
must add up to 2 (to get this, take the definition of a perfect number,
1401:
perfect numbers, nor whether there are infinitely many Mersenne primes.
9685:
9344:
7712:
7499:
7316:
7281:
7246:
6697:
6020:
5902:
3298:
478:
58:
7182:
7165:
6735:
6733:
6731:
6391:
5617:
The Development of Arabic Mathematics: Between Arithmetic and Algebra
3420:
The second largest prime factor is greater than 10, and is less than
495:
446:
192:
7490:
7471:
7268:
Hornfeck, B (1955). "Zur Dichte der Menge der vollkommenen zahlen".
7238:
4198:) = (1, ..., 1, 2, ..., 2) with
7413:
for a translation and discussion of this proposition and its proof.
7093:
6593:
6211:
3962:{\displaystyle qp_{1}p_{2}p_{3}\cdots p_{k}<2N^{\frac {17}{26}}}
7983:
6294:
Bibby, Sean; Vyncke, Pieter; Zelinsky, Joshua (23 November 2021).
5055:
4850:
From these two results it follows that every perfect number is an
3457:
The third largest prime factor is greater than 100, and less than
1302:
36:
7610:
5603:
Society of Biblical Literature National Meeting, Atlanta, Georgia
4082:
Furthermore, several minor results are known about the exponents
3601:{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}},}
5716:. Washington: Mathematical Association of America. p. 132.
5686:
Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning
5619:(Dordrecht: Kluwer Academic Publishers, 1994), pp. 328â329.
9971:
9935:
9899:
9863:
9823:
9448:
9337:
9063:
8978:
8933:
8810:
8500:
8447:
8399:
8333:
8285:
8223:
8127:
8088:
7668:
6442:"On the Total Number of Prime Factors of an Odd Perfect Number"
6372:"Odd perfect numbers have at least nine distinct prime factors"
7581:
7374:
Number Theory: An Introduction to Pure and Applied Mathematics
6948:"A new result concerning the structure of odd perfect numbers"
6329:"Odd perfect numbers, Diophantine equations, and upper bounds"
5924:"Mathematicians Open a New Front on an Ancient Number Problem"
7360:. Washington: Carnegie Institution of Washington. p. 25.
6296:"On the Third Largest Prime Divisor of an Odd Perfect Number"
5672:. Washington: Carnegie Institution of Washington. p. 10.
174:
7624:
6897:
6895:
5864:. Washington: Carnegie Institution of Washington. p. 6.
5521:. Washington: Carnegie Institution of Washington. p. 4.
5233:
1572:. Furthermore, each even perfect number except for 6 is the
7613:
6904:"Extensions of some results concerning odd perfect numbers"
6740:
Fletcher, S. Adam; Nielsen, Pace P.; Ochem, Pascal (2012).
5809:
5475:
5450:
2891:
every even perfect number is represented in binary form as
1369:
1290:
will yield all the even perfect numbers. Thus, there is a
6539:"On the number of prime factors of an odd perfect number"
5713:
Mathematical Treks: From Surreal Numbers to Magic Circles
4290:) ≠ (1, ..., 1, 3), (1, ..., 1, 5), (1, ..., 1, 6).
41:
Illustration of the perfect number status of the number 6
7472:"A Lower Bound for the set of odd Perfect Prime Numbers"
6800:
Journal of the Australian Mathematical Society, Series A
6665:
Suryanarayana, D. (1963). "On Odd Perfect Numbers II".
5948:
5946:
5944:
1227:
is prime (though he stated this somewhat differently),
6070:"Odd perfect numbers have a prime factor exceeding 10"
4264:{\displaystyle (t-1)/4\leq u\leq 2t+{\sqrt {\alpha }}}
1580:
1502:(and hence equal to the sum of the integers from 1 to
1050:
itself be prime. However, not all numbers of the form
7204:, John Wiley & Sons, Section 2.3, Exercise 2(6),
7003:
5408:
5328:
5230:
5002:
4971:
4931:
4873:
4740:
4559:
4507:
4380:
4329:
4212:
3977:
3891:
3775:
3703:
3657:
3525:
3463:
3426:
3387:
2910:
2839:
2777:
2714:
2483:
1714:
1666:
1626:
1578:
1541:
1508:
1465:
1410:
1311:
1241:
1200:
1145:
1101:
1056:
1019:
974:
667:
601:
559:
504:
410:
355:
300:
267:
244:
201:
138:
93:
83:
The sum of proper divisors of a number is called its
7515:
Prime Numbers and Computer Methods for Factorisation
6475:"Improved upper bounds for odd multiperfect numbers"
9795:
9749:
9709:
9660:
9634:
9567:
9551:
9530:
9497:
9462:
9302:
9269:
9226:
9203:
9080:
8768:
8759:
8737:
8694:
8656:
8647:
8580:
8522:
8513:
8026:
7982:
7943:
7892:
7826:
7750:
7700:
7201:
Computational Number Theory and Modern Cryptography
6796:"On the largest component of an odd perfect number"
5735:
5733:
7145:Makowski, A. (1962). "Remark on perfect numbers".
7029:
5430:
5347:
5240:
5021:
4984:
4950:
4914:
4830:
4723:
4538:
4431:
4366:
4263:
4068:
3961:
3875:
3760:
3681:
3600:
3486:
3446:
3410:
3266:
2883:
2818:
2755:
2626:
2467:
1698:
1652:
1608:
1560:
1527:
1490:
1451:
1352:
1282:
1219:
1186:
1120:
1075:
1038:
993:
958:
645:
578:
545:
429:
396:
306:
286:
250:
230:
166:This definition is ancient, appearing as early as
151:
124:
3244:
3243:
3154:
3153:
6952:Proceedings of the American Mathematical Society
6667:Proceedings of the American Mathematical Society
6646:Cohen, Graeme (1978). "On odd perfect numbers".
6435:
6433:
5689:. Oxford: Oxford University Press. p. 360.
2477:Even perfect numbers (except 6) are of the form
57:that is equal to the sum of its positive proper
7419:Journal fĂŒr die Reine und Angewandte Mathematik
7166:"On a remark of Makowski about perfect numbers"
6849:Journal fĂŒr die reine und angewandte Mathematik
6479:Bulletin of the Australian Mathematical Society
5040:of every even perfect number other than 6 is 1.
4451:
27:Integer equal to the sum of its proper divisors
7510:, Vol. 154, Amsterdam, 1982, pp. 141â157.
7233:(497). The Mathematical Association: 262â263.
8100:
7680:
6995:McDaniel, Wayne L.; Hagis, Peter Jr. (1975).
6946:Hagis, Peter Jr.; McDaniel, Wayne L. (1972).
4167:+1 have a prime factor in a given finite set
195:also proved a formation rule (IX.36) whereby
8:
7614:sequence A000396 (Perfect numbers)
5284:List of Mersenne primes and perfect numbers
5164:, and where it is greater than the number,
4831:{\displaystyle 1/28+1/14+1/7+1/4+1/2+1/1=2}
9968:
9932:
9896:
9860:
9820:
9494:
9459:
9445:
9334:
9077:
9060:
8975:
8930:
8807:
8765:
8653:
8519:
8510:
8497:
8444:
8401:Possessing a specific set of other numbers
8396:
8330:
8282:
8220:
8124:
8107:
8093:
8085:
7687:
7673:
7665:
7521:SĂĄndor, Jozsef; Crstici, Borislav (2004).
5443:= 11, 23, 83, 131, 179, 191, 239, 251, ...
5180:, and larger cycles of numbers are called
3761:{\displaystyle N<2^{(4^{k+1}-2^{k+1})}}
3691:At least one of the prime powers dividing
486:Are there infinitely many perfect numbers?
7489:
7181:
7092:
7018:
7008:
7002:
6963:
6811:
6762:
6592:
6556:
6490:
6390:
6347:
6271:
6210:
6174:
6096:
5975:
5955:"Odd perfect numbers are greater than 10"
5413:
5407:
5333:
5327:
5232:
5231:
5229:
5009:
5001:
4975:
4970:
4936:
4930:
4897:
4878:
4872:
4814:
4800:
4786:
4772:
4758:
4744:
4739:
4697:
4664:
4651:
4638:
4625:
4612:
4599:
4586:
4573:
4560:
4558:
4512:
4506:
4475:The only even perfect number of the form
4404:
4396:
4391:
4379:
4343:
4328:
4254:
4228:
4211:
4046:
4037:
4020:
4011:
4000:
3991:
3978:
3976:
3948:
3932:
3919:
3909:
3899:
3890:
3852:
3843:
3821:
3805:
3789:
3774:
3741:
3722:
3714:
3702:
3658:
3656:
3587:
3579:
3574:
3559:
3551:
3546:
3536:
3524:
3474:
3464:
3462:
3437:
3427:
3425:
3398:
3388:
3386:
3254:
3237:
3224:
3211:
3198:
3185:
3172:
3159:
3142:
3128:
3113:
3100:
3087:
3074:
3061:
3046:
3032:
3017:
3004:
2991:
2976:
2962:
2947:
2934:
2919:
2911:
2909:
2863:
2844:
2838:
2801:
2782:
2776:
2738:
2719:
2713:
2614:
2599:
2591:
2551:
2526:
2516:
2493:
2488:
2482:
2455:
2442:
2429:
2410:
2397:
2384:
2306:
2293:
2269:
2256:
2243:
2230:
2217:
2204:
2191:
2178:
2100:
2087:
2063:
2050:
2037:
2024:
1946:
1933:
1909:
1896:
1818:
1805:
1746:
1733:
1715:
1713:
1671:
1665:
1631:
1625:
1587:
1579:
1577:
1546:
1540:
1513:
1507:
1473:
1464:
1434:
1415:
1409:
1335:
1316:
1310:
1265:
1246:
1240:
1205:
1199:
1169:
1150:
1144:
1106:
1100:
1061:
1055:
1024:
1018:
979:
973:
919:
906:
848:
835:
777:
764:
706:
693:
668:
666:
625:
606:
600:
564:
558:
528:
509:
503:
415:
409:
379:
360:
354:
299:
272:
266:
243:
220:
200:
143:
137:
98:
92:
7357:History of the Theory of Numbers, Vol. I
6508:"An upper bound for odd perfect numbers"
5876:"The oldest open problem in mathematics"
5861:History of the Theory of Numbers, Vol. I
5669:History of the Theory of Numbers, Vol. I
5518:History of the Theory of Numbers, Vol. I
4961:The number of perfect numbers less than
4487:
4480:
4432:{\displaystyle N<2^{4^{2e^{2}+8e+3}}}
2826:for odd integer (not necessarily prime)
7529:. Dordrecht: Kluwer Academic. pp.
7133:Compte Rendu de l'Association Française
6742:"Sieve methods for odd perfect numbers"
6615:Pomerance, Carl; Luca, Florian (2010).
5792:MacTutor History of Mathematics Archive
5467:
5315:
4321:cannot be 3, 5, 24, 6, 8, 11, 14 or 18.
3507:has at least 12 distinct prime factors.
3343:must satisfy the following conditions:
3310:(more unsolved problems in mathematics)
1660:odd cubes (odd cubes up to the cube of
490:(more unsolved problems in mathematics)
7582:Perfect, amicable and sociable numbers
7508:Computational Methods in Number Theory
7030:{\displaystyle p^{\alpha }M^{2\beta }}
6902:Cohen, G. L.; Williams, R. J. (1985).
6199:International Journal of Number Theory
6121:International Journal of Number Theory
6044:"On the Form of an Odd Perfect Number"
5759:
4151:5), then the smallest prime factor of
2708:. This works with all perfect numbers
2636:with each resulting triangular number
437:is prime. He seems to be unaware that
5191:By definition, a perfect number is a
3288:
1620:and is equal to the sum of the first
1609:{\displaystyle {\tfrac {2^{p}+1}{3}}}
1005:, after the seventeenth-century monk
7:
7625:Great Internet Mersenne Prime Search
7587:Perfect numbers â History and Theory
6617:"On the radical of a perfect number"
6572:Graeme Clayton, Cody Hansen (2023).
6537:Ochem, Pascal; Rao, Michaël (2014).
6473:Chen, Yong-Gao; Tang, Cui-E (2014).
5953:Ochem, Pascal; Rao, Michaël (2012).
5746:Great Internet Mersenne Prime Search
4996:> 0 is a constant. In fact it is
4497:of the divisors of a perfect number
4171:, then the smallest prime factor of
3283:Every even perfect number is also a
3276:Thus every even perfect number is a
1699:{\displaystyle 2^{\frac {p+1}{2}}-1}
7695:Divisibility-based sets of integers
5394:is congruent to 1 or 7 mod 8, then
553:is an even perfect number whenever
238:is an even perfect number whenever
64:The first four perfect numbers are
10039:Unsolved problems in number theory
6440:Zelinsky, Joshua (3 August 2021).
5922:Nadis, Steve (10 September 2020).
3306:Are there any odd perfect numbers?
1653:{\displaystyle 2^{\frac {p-1}{2}}}
1459:, each even perfect number is the
1046:to be prime, it is necessary that
25:
7733:Fundamental theorem of arithmetic
6965:10.1090/S0002-9939-1972-0292740-5
5787:"Abu Ali al-Hasan ibn al-Haytham"
5639:History of Mathematics: Volume II
4859:The even perfect numbers are not
4539:{\displaystyle \sigma _{1}(n)=2n}
4367:{\displaystyle k\leq 2e^{2}+8e+2}
3682:{\displaystyle {\frac {k-1}{2}}.}
3381:is greater than 10 and less than
2884:{\displaystyle 2^{p-1}(2^{p}-1),}
1139:perfect numbers were of the form
646:{\displaystyle 2^{p-1}(2^{p}-1),}
125:{\displaystyle \sigma _{1}(n)=2n}
10002:
9610:Perfect digit-to-digit invariant
7741:
4915:{\displaystyle 2^{n-1}(2^{n}+1)}
3634:are distinct odd primes (Euler).
2819:{\displaystyle 2^{m-1}(2^{m}-1)}
2756:{\displaystyle 2^{p-1}(2^{p}-1)}
1452:{\displaystyle 2^{p-1}(2^{p}-1)}
1353:{\displaystyle 2^{p-1}(2^{p}-1)}
1283:{\displaystyle 2^{p-1}(2^{p}-1)}
1187:{\displaystyle 2^{n-1}(2^{n}-1)}
1135:had stated (without proof) that
546:{\displaystyle 2^{p-1}(2^{p}-1)}
397:{\displaystyle 2^{n-1}(2^{n}-1)}
7409:, Book IX, Proposition 36. See
6764:10.1090/S0025-5718-2011-02576-7
6621:New York Journal of Mathematics
6558:10.1090/S0025-5718-2013-02776-7
6349:10.1090/S0025-5718-2015-02941-X
6051:Australian Mathematical Gazette
5977:10.1090/S0025-5718-2012-02563-4
5642:. New York: Dover. p. 21.
3301:Unsolved problem in mathematics
1393:with 49,724,095 digits). It is
901:
830:
759:
688:
481:Unsolved problem in mathematics
7341:Texeira J. VIII (1886), 11â16.
7135:(Toulouse, 1887), pp. 164â168.
5241:{\displaystyle {\mathcal {S}}}
5022:{\displaystyle o({\sqrt {n}})}
5016:
5006:
4909:
4890:
4524:
4518:
4225:
4213:
3753:
3715:
2875:
2856:
2813:
2794:
2750:
2731:
2611:
2592:
2563:
2544:
2538:
2519:
2318:
2299:
2112:
2093:
1958:
1939:
1830:
1811:
1758:
1739:
1485:
1466:
1446:
1427:
1347:
1328:
1277:
1258:
1181:
1162:
1128:is prime for only 48 of them.
931:
912:
860:
841:
789:
770:
718:
699:
637:
618:
540:
521:
391:
372:
349:perfect number is of the form
217:
205:
110:
104:
1:
8449:Expressible via specific sums
7339:Note sur les nombres parfaits
6409:10.1090/S0025-5718-07-01990-4
6273:10.1090/S0025-5718-99-01127-8
6176:10.1090/S0025-5718-99-01126-6
6098:10.1090/S0025-5718-08-02050-9
5168:. These terms, together with
3647:The smallest prime factor of
7525:Handbook of number theory II
4985:{\displaystyle c{\sqrt {n}}}
4847:cannot be a perfect square.
3487:{\displaystyle {\sqrt{2N}}.}
3411:{\displaystyle {\sqrt{3N}}.}
3377:The largest prime factor of
1301:An exhaustive search by the
173:(VII.22) where it is called
9538:Multiplicative digital root
7569:Encyclopedia of Mathematics
5629:Bayerische Staatsbibliothek
5537:www-groups.dcs.st-and.ac.uk
5197:restricted divisor function
4922:formed as the product of a
4546:, and divide both sides by
4469:strong law of small numbers
4155:must lie between 10 and 10.
3447:{\displaystyle {\sqrt{2N}}}
1404:As well as having the form
152:{\displaystyle \sigma _{1}}
10065:
7477:Mathematics of Computation
7164:Gallardo, Luis H. (2010).
6750:Mathematics of Computation
6544:Mathematics of Computation
6379:Mathematics of Computation
6336:Mathematics of Computation
6252:Mathematics of Computation
6155:Mathematics of Computation
6077:Mathematics of Computation
5963:Mathematics of Computation
5766:: CS1 maint: url-status (
5589:Rogers, Justin M. (2015).
5566:based on a Mersenne prime.
5559:Introduction to Arithmetic
968:Prime numbers of the form
472:
175:
29:
9998:
9981:
9967:
9945:
9931:
9909:
9895:
9873:
9859:
9832:
9819:
9615:Perfect digital invariant
9458:
9444:
9352:
9333:
9190:Superior highly composite
9076:
9059:
8987:
8974:
8942:
8929:
8817:
8806:
8509:
8496:
8454:
8443:
8406:
8395:
8343:
8329:
8292:
8281:
8234:
8219:
8137:
8123:
7930:Superior highly composite
7739:
6862:10.1515/crll.1950.188.129
6813:10.1017/S1446788700028251
6506:Nielsen, Pace P. (2003).
6492:10.1017/S0004972713000488
6370:Nielsen, Pace P. (2007).
6327:Nielsen, Pace P. (2015).
6221:10.1142/S1793042119500659
6133:10.1142/S1793042112500935
6068:Goto, T; Ohno, Y (2008).
6009:Mathematische Zeitschrift
5741:"GIMPS Milestones Report"
5579:edition: vol. 385, 58â61.
5114:superior highly composite
1618:centered nonagonal number
1491:{\displaystyle (2^{p}-1)}
1292:one-to-one correspondence
1013:and perfect numbers. For
9228:Euler's totient function
9012:EulerâJacobi pseudoprime
8287:Other polynomial numbers
7827:Constrained divisor sums
7431:10.1515/crll.1941.183.98
7227:The Mathematical Gazette
5797:University of St Andrews
5431:{\displaystyle 2^{p}-1,}
5172:itself, come from Greek
4158:More generally, if all 2
3356:is not divisible by 105.
3330:harmonic divisor numbers
1235:proved that the formula
1087:are prime; for example,
341:, and the mathematician
231:{\displaystyle q(q+1)/2}
161:sum-of-divisors function
9042:SomerâLucas pseudoprime
9032:LucasâCarmichael number
8867:Lazy caterer's sequence
7639:"8128: Perfect Numbers"
7633:, math forum at Drexel.
7448:S.-B. Bayer. Akad. Wiss
7081:Colloquium Mathematicum
5348:{\displaystyle 2^{p}-1}
5304:Harmonic divisor number
5289:Multiply perfect number
4951:{\displaystyle 2^{n}+1}
3339:Any odd perfect number
1561:{\displaystyle 2^{p-1}}
1528:{\displaystyle 2^{p}-1}
1220:{\displaystyle 2^{n}-1}
1121:{\displaystyle 2^{p}-1}
1076:{\displaystyle 2^{p}-1}
1039:{\displaystyle 2^{p}-1}
994:{\displaystyle 2^{p}-1}
579:{\displaystyle 2^{p}-1}
430:{\displaystyle 2^{n}-1}
318:. Two millennia later,
287:{\displaystyle 2^{p}-1}
30:For the 2012 film, see
8917:WedderburnâEtherington
8317:Lucky numbers of Euler
7031:
5438:which is the case for
5432:
5368:2 â 1 = 2047 = 23 Ă 89
5349:
5299:Unitary perfect number
5242:
5153:
5063:of numbers under 100:
5023:
4986:
4952:
4916:
4832:
4725:
4540:
4479: + 1 is 28 (
4455:
4433:
4368:
4265:
4070:
3963:
3877:
3762:
3683:
3602:
3488:
3448:
3412:
3268:
2885:
2820:
2757:
2628:
2469:
1700:
1654:
1610:
1562:
1529:
1492:
1453:
1354:
1284:
1221:
1188:
1122:
1089:2 â 1 = 2047 = 23 Ă 89
1077:
1040:
995:
960:
647:
580:
547:
431:
398:
314:âwhat is now called a
308:
288:
252:
232:
153:
126:
42:
9205:Prime omega functions
9022:Frobenius pseudoprime
8812:Combinatorial numbers
8681:Centered dodecahedral
8474:Primary pseudoperfect
7708:Integer factorization
7371:Redmond, Don (1996).
7198:Yan, Song Y. (2012),
7103:10.4064/cm7339-3-2018
7032:
6794:Cohen, G. L. (1987).
6686:Archiv der Mathematik
6243:Iannucci, DE (2000).
6146:Iannucci, DE (1999).
5594:Commentary on Genesis
5433:
5350:
5248:-perfect numbers, or
5243:
5059:
5024:
4987:
4953:
4917:
4852:Ore's harmonic number
4833:
4726:
4541:
4434:
4369:
4266:
4071:
3964:
3878:
3763:
3684:
3603:
3489:
3449:
3413:
3269:
2886:
2833:Owing to their form,
2821:
2758:
2629:
2470:
1701:
1655:
1611:
1563:
1530:
1493:
1454:
1355:
1285:
1222:
1189:
1123:
1078:
1041:
996:
961:
648:
581:
548:
432:
399:
309:
294:for positive integer
289:
253:
233:
154:
127:
40:
32:Perfect Number (film)
9664:-composition related
9464:Arithmetic functions
9066:Arithmetic functions
9002:Elliptic pseudoprime
8686:Centered icosahedral
8666:Centered tetrahedral
7411:D.E. Joyce's website
7001:
6844:Kanold, Hans-Joachim
5783:Robertson, Edmund F.
5710:Peterson, I (2002).
5683:Pickover, C (2001).
5406:
5402:will be a factor of
5376:Sophie Germain prime
5326:
5294:Superperfect numbers
5228:
5047:perfect number is 6.
5000:
4969:
4929:
4871:
4738:
4557:
4505:
4378:
4327:
4210:
3975:
3889:
3773:
3701:
3655:
3523:
3461:
3424:
3385:
2908:
2902:zeros; for example:
2837:
2775:
2771:numbers of the form
2712:
2481:
1712:
1664:
1624:
1576:
1539:
1506:
1463:
1408:
1309:
1296:EuclidâEuler theorem
1239:
1198:
1143:
1099:
1054:
1017:
972:
665:
599:
557:
502:
475:EuclidâEuler theorem
469:Even perfect numbers
408:
353:
324:EuclidâEuler theorem
298:
265:
242:
199:
136:
91:
9590:Kaprekar's constant
9110:Colossally abundant
8997:Catalan pseudoprime
8897:SchröderâHipparchus
8676:Centered octahedral
8552:Centered heptagonal
8542:Centered pentagonal
8532:Centered triangular
8132:and related numbers
7920:Colossally abundant
7751:Factorization forms
7517:, Birkhauser, 1985.
7045:Fibonacci Quarterly
6912:Fibonacci Quarterly
6648:Fibonacci Quarterly
6401:2007MaCom..76.2109N
6264:2000MaCom..69..867I
6167:1999MaCom..68.1749I
6089:2008MaCom..77.1859G
6042:Roberts, T (2008).
5781:O'Connor, John J.;
5577:Sources Chrétiennes
5274:Hyperperfect number
5110:Colossally abundant
4861:trapezoidal numbers
3640:⥠α ⥠1 (
3594:
3566:
3370:⥠117 (mod 468) or
3295:Odd perfect numbers
2767:and, in fact, with
443:Philo of Alexandria
10008:Mathematics portal
9950:Aronson's sequence
9696:SmarandacheâWellin
9453:-dependent numbers
9160:Primitive abundant
9047:Strong pseudoprime
9037:Perrin pseudoprime
9017:Fermat pseudoprime
8957:Wolstenholme prime
8781:Squared triangular
8567:Centered decagonal
8562:Centered nonagonal
8557:Centered octagonal
8547:Centered hexagonal
7905:Primitive abundant
7893:With many divisors
7594:Weisstein, Eric W.
7470:Hagis, P. (1973).
7317:10.1007/BF01350108
7282:10.1007/BF01901120
7027:
6757:(279): 1753?1776.
6698:10.1007/BF01220877
6551:(289): 2435â2439.
6385:(260): 2109â2126.
6342:(295): 2549â2567.
6161:(228): 1749â1760.
6083:(263): 1859â1868.
6021:10.1007/BF02230691
5970:(279): 1869â1877.
5909:2006-12-29 at the
5832:Weisstein, Eric W.
5634:David Eugene Smith
5428:
5386:is also primeâand
5345:
5257:semiperfect number
5238:
5154:
5079:Primitive abundant
5019:
4982:
4948:
4912:
4865:triangular numbers
4828:
4721:
4536:
4429:
4364:
4261:
4066:
3959:
3873:
3758:
3679:
3598:
3570:
3542:
3484:
3444:
3408:
3326:heuristic argument
3264:
3262:
2881:
2816:
2753:
2698:8 + 1 + 2 + 8 = 19
2624:
2465:
2463:
1696:
1650:
1606:
1604:
1558:
1525:
1488:
1449:
1397:whether there are
1350:
1280:
1217:
1184:
1118:
1095:up to 68,874,199,
1073:
1036:
991:
956:
954:
643:
576:
543:
427:
394:
304:
284:
248:
228:
149:
122:
43:
10034:Integer sequences
10016:
10015:
9994:
9993:
9963:
9962:
9927:
9926:
9891:
9890:
9855:
9854:
9815:
9814:
9811:
9810:
9630:
9629:
9440:
9439:
9329:
9328:
9325:
9324:
9271:Aliquot sequences
9082:Divisor functions
9055:
9054:
9027:Lucas pseudoprime
9007:Euler pseudoprime
8992:Carmichael number
8970:
8969:
8925:
8924:
8802:
8801:
8798:
8797:
8794:
8793:
8755:
8754:
8643:
8642:
8600:Square triangular
8492:
8491:
8439:
8438:
8391:
8390:
8325:
8324:
8277:
8276:
8215:
8214:
8082:
8081:
5631:, Clm 14908. See
5564:triangular number
5533:"Perfect numbers"
5498:Caldwell, Chris,
5355:are congruent to
5250:Granville numbers
5031:little-o notation
5014:
4980:
4713:
4692:
4659:
4646:
4633:
4620:
4607:
4594:
4581:
4568:
4281:, ...,
4259:
4189:, ...,
4089:, ...,
4052:
4026:
4006:
3986:
3956:
3871:
3674:
3625:, ...,
3479:
3442:
3403:
3334:Descartes numbers
3278:pernicious number
2895:ones followed by
2570:
1687:
1647:
1603:
1500:triangular number
459:Ismail ibn Fallƫs
451:Didymus the Blind
339:Greek mathematics
307:{\displaystyle p}
251:{\displaystyle q}
16:(Redirected from
10056:
10029:Divisor function
10006:
9969:
9938:Natural language
9933:
9897:
9865:Generated via a
9861:
9821:
9726:Digit-reassembly
9691:Self-descriptive
9495:
9460:
9446:
9397:LucasâCarmichael
9387:Harmonic divisor
9335:
9261:Sparsely totient
9236:Highly cototient
9145:Multiply perfect
9135:Highly composite
9078:
9061:
8976:
8931:
8912:Telephone number
8808:
8766:
8747:Square pyramidal
8729:Stella octangula
8654:
8520:
8511:
8503:Figurate numbers
8498:
8445:
8397:
8331:
8283:
8221:
8125:
8109:
8102:
8095:
8086:
8059:Harmonic divisor
7945:Aliquot sequence
7925:Highly composite
7849:Multiply perfect
7745:
7723:Divisor function
7689:
7682:
7675:
7666:
7661:
7659:
7658:
7649:. Archived from
7612:
7607:
7606:
7597:"Perfect Number"
7577:
7564:"Perfect number"
7552:
7528:
7503:
7493:
7484:(124): 951â953.
7455:
7442:
7390:
7388:
7368:
7362:
7361:
7348:
7342:
7335:
7329:
7328:
7300:
7294:
7293:
7265:
7259:
7258:
7222:
7216:
7214:
7195:
7189:
7187:
7185:
7161:
7155:
7154:
7142:
7136:
7129:
7123:
7122:
7096:
7076:
7070:
7069:
7041:
7036:
7034:
7033:
7028:
7026:
7025:
7013:
7012:
6992:
6986:
6985:
6967:
6943:
6937:
6936:
6908:
6899:
6890:
6889:
6840:
6834:
6833:
6815:
6791:
6785:
6784:
6766:
6746:
6737:
6726:
6725:
6681:
6675:
6674:
6662:
6656:
6655:
6643:
6637:
6636:
6634:
6632:
6612:
6606:
6605:
6603:
6601:
6596:
6578:
6569:
6563:
6562:
6560:
6534:
6528:
6527:
6525:
6523:
6503:
6497:
6496:
6494:
6470:
6464:
6463:
6461:
6459:
6446:
6437:
6428:
6427:
6425:
6423:
6394:
6376:
6367:
6361:
6360:
6358:
6356:
6351:
6333:
6324:
6318:
6317:
6315:
6313:
6300:
6291:
6285:
6284:
6282:
6280:
6275:
6258:(230): 867â879.
6249:
6240:
6234:
6232:
6214:
6205:(6): 1183â1189.
6194:
6188:
6187:
6185:
6183:
6178:
6152:
6143:
6137:
6136:
6127:(6): 1537â1540.
6116:
6110:
6109:
6107:
6105:
6100:
6074:
6065:
6059:
6058:
6048:
6039:
6033:
6032:
6004:
5998:
5997:
5979:
5959:
5950:
5939:
5938:
5936:
5934:
5919:
5913:
5900:
5894:
5893:
5891:
5889:
5880:
5872:
5866:
5865:
5852:
5846:
5845:
5844:
5835:"Perfect Number"
5827:
5821:
5820:
5818:
5817:
5806:
5800:
5799:
5778:
5772:
5771:
5765:
5757:
5755:
5753:
5737:
5728:
5727:
5707:
5701:
5700:
5680:
5674:
5673:
5660:
5654:
5653:
5626:
5620:
5613:
5607:
5606:
5600:
5586:
5580:
5573:
5567:
5554:
5548:
5547:
5545:
5543:
5529:
5523:
5522:
5509:
5503:
5496:
5490:
5489:
5487:
5486:
5476:"A000396 - OEIS"
5472:
5455:
5453:
5444:
5442:
5437:
5435:
5434:
5429:
5418:
5417:
5401:
5393:
5385:
5373:
5369:
5365:
5354:
5352:
5351:
5346:
5338:
5337:
5320:
5247:
5245:
5244:
5239:
5237:
5236:
5222:aliquot sequence
5219:
5186:practical number
5147:
5138:
5129:
5120:
5107:
5101:highly composite
5094:
5085:
5076:
5067:
5052:Related concepts
5028:
5026:
5025:
5020:
5015:
5010:
4991:
4989:
4988:
4983:
4981:
4976:
4957:
4955:
4954:
4949:
4941:
4940:
4921:
4919:
4918:
4913:
4902:
4901:
4889:
4888:
4837:
4835:
4834:
4829:
4818:
4804:
4790:
4776:
4762:
4748:
4734:For 28, we have
4730:
4728:
4727:
4722:
4714:
4709:
4698:
4693:
4688:
4665:
4660:
4652:
4647:
4639:
4634:
4626:
4621:
4613:
4608:
4600:
4595:
4587:
4582:
4574:
4569:
4561:
4545:
4543:
4542:
4537:
4517:
4516:
4438:
4436:
4435:
4430:
4428:
4427:
4426:
4425:
4409:
4408:
4373:
4371:
4370:
4365:
4348:
4347:
4314:
4270:
4268:
4267:
4262:
4260:
4255:
4232:
4143: ⥠1 (
4127: ⥠2 (
4111: ⥠1 (
4075:
4073:
4072:
4067:
4053:
4051:
4050:
4038:
4027:
4025:
4024:
4012:
4007:
4005:
4004:
3992:
3987:
3979:
3968:
3966:
3965:
3960:
3958:
3957:
3949:
3937:
3936:
3924:
3923:
3914:
3913:
3904:
3903:
3882:
3880:
3879:
3874:
3872:
3867:
3853:
3848:
3847:
3826:
3825:
3810:
3809:
3794:
3793:
3767:
3765:
3764:
3759:
3757:
3756:
3752:
3751:
3733:
3732:
3688:
3686:
3685:
3680:
3675:
3670:
3659:
3607:
3605:
3604:
3599:
3593:
3592:
3591:
3578:
3565:
3564:
3563:
3550:
3541:
3540:
3493:
3491:
3490:
3485:
3480:
3478:
3473:
3465:
3453:
3451:
3450:
3445:
3443:
3441:
3436:
3428:
3417:
3415:
3414:
3409:
3404:
3402:
3397:
3389:
3366:⥠1 (mod 12) or
3324:has presented a
3302:
3289:Related concepts
3285:practical number
3273:
3271:
3270:
3265:
3263:
3259:
3258:
3242:
3241:
3229:
3228:
3216:
3215:
3203:
3202:
3190:
3189:
3177:
3176:
3164:
3163:
3147:
3146:
3133:
3132:
3118:
3117:
3105:
3104:
3092:
3091:
3079:
3078:
3066:
3065:
3051:
3050:
3037:
3036:
3022:
3021:
3009:
3008:
2996:
2995:
2981:
2980:
2967:
2966:
2952:
2951:
2939:
2938:
2924:
2923:
2901:
2894:
2890:
2888:
2887:
2882:
2868:
2867:
2855:
2854:
2829:
2825:
2823:
2822:
2817:
2806:
2805:
2793:
2792:
2766:
2762:
2760:
2759:
2754:
2743:
2742:
2730:
2729:
2707:
2703:
2699:
2691:
2683:
2675:
2667:
2659:
2651:
2643:
2633:
2631:
2630:
2625:
2623:
2622:
2618:
2604:
2603:
2571:
2566:
2556:
2555:
2531:
2530:
2517:
2506:
2505:
2498:
2497:
2474:
2472:
2471:
2466:
2464:
2460:
2459:
2447:
2446:
2434:
2433:
2415:
2414:
2402:
2401:
2389:
2388:
2373:
2372:
2371:
2322:
2311:
2310:
2298:
2297:
2274:
2273:
2261:
2260:
2248:
2247:
2235:
2234:
2222:
2221:
2209:
2208:
2196:
2195:
2183:
2182:
2167:
2166:
2165:
2116:
2105:
2104:
2092:
2091:
2068:
2067:
2055:
2054:
2042:
2041:
2029:
2028:
2013:
2012:
2011:
1962:
1951:
1950:
1938:
1937:
1914:
1913:
1901:
1900:
1885:
1884:
1883:
1834:
1823:
1822:
1810:
1809:
1762:
1751:
1750:
1738:
1737:
1705:
1703:
1702:
1697:
1689:
1688:
1683:
1672:
1659:
1657:
1656:
1651:
1649:
1648:
1643:
1632:
1615:
1613:
1612:
1607:
1605:
1599:
1592:
1591:
1581:
1570:hexagonal number
1567:
1565:
1564:
1559:
1557:
1556:
1534:
1532:
1531:
1526:
1518:
1517:
1497:
1495:
1494:
1489:
1478:
1477:
1458:
1456:
1455:
1450:
1439:
1438:
1426:
1425:
1392:
1388:
1384:
1372:
1366:
1359:
1357:
1356:
1351:
1340:
1339:
1327:
1326:
1289:
1287:
1286:
1281:
1270:
1269:
1257:
1256:
1226:
1224:
1223:
1218:
1210:
1209:
1193:
1191:
1190:
1185:
1174:
1173:
1161:
1160:
1127:
1125:
1124:
1119:
1111:
1110:
1094:
1090:
1086:
1082:
1080:
1079:
1074:
1066:
1065:
1049:
1045:
1043:
1042:
1037:
1029:
1028:
1000:
998:
997:
992:
984:
983:
965:
963:
962:
957:
955:
924:
923:
911:
910:
853:
852:
840:
839:
782:
781:
769:
768:
711:
710:
698:
697:
656:
652:
650:
649:
644:
630:
629:
617:
616:
592:, Prop. IX.36).
585:
583:
582:
577:
569:
568:
552:
550:
549:
544:
533:
532:
520:
519:
482:
440:
436:
434:
433:
428:
420:
419:
403:
401:
400:
395:
384:
383:
371:
370:
313:
311:
310:
305:
293:
291:
290:
285:
277:
276:
257:
255:
254:
249:
237:
235:
234:
229:
224:
178:
177:
158:
156:
155:
150:
148:
147:
131:
129:
128:
123:
103:
102:
55:positive integer
21:
10064:
10063:
10059:
10058:
10057:
10055:
10054:
10053:
10049:Perfect numbers
10044:Mersenne primes
10019:
10018:
10017:
10012:
9990:
9986:Strobogrammatic
9977:
9959:
9941:
9923:
9905:
9887:
9869:
9851:
9828:
9807:
9791:
9750:Divisor-related
9745:
9705:
9656:
9626:
9563:
9547:
9526:
9493:
9466:
9454:
9436:
9348:
9347:related numbers
9321:
9298:
9265:
9256:Perfect totient
9222:
9199:
9130:Highly abundant
9072:
9051:
8983:
8966:
8938:
8921:
8907:Stirling second
8813:
8790:
8751:
8733:
8690:
8639:
8576:
8537:Centered square
8505:
8488:
8450:
8435:
8402:
8387:
8339:
8338:defined numbers
8321:
8288:
8273:
8244:Double Mersenne
8230:
8211:
8133:
8119:
8117:natural numbers
8113:
8083:
8078:
8022:
7978:
7939:
7910:Highly abundant
7888:
7869:Unitary perfect
7822:
7746:
7737:
7718:Unitary divisor
7696:
7693:
7656:
7654:
7637:Grimes, James.
7636:
7631:Perfect Numbers
7592:
7591:
7562:
7559:
7541:
7520:
7491:10.2307/2005530
7469:
7463:
7461:Further reading
7458:
7445:
7425:(183): 98â109.
7416:
7399:
7394:
7393:
7385:
7370:
7369:
7365:
7350:
7349:
7345:
7336:
7332:
7302:
7301:
7297:
7267:
7266:
7262:
7239:10.2307/3619053
7224:
7223:
7219:
7212:
7197:
7196:
7192:
7163:
7162:
7158:
7144:
7143:
7139:
7130:
7126:
7078:
7077:
7073:
7039:
7014:
7004:
6999:
6998:
6994:
6993:
6989:
6945:
6944:
6940:
6906:
6901:
6900:
6893:
6842:
6841:
6837:
6793:
6792:
6788:
6744:
6739:
6738:
6729:
6683:
6682:
6678:
6664:
6663:
6659:
6645:
6644:
6640:
6630:
6628:
6614:
6613:
6609:
6599:
6597:
6576:
6571:
6570:
6566:
6536:
6535:
6531:
6521:
6519:
6505:
6504:
6500:
6472:
6471:
6467:
6457:
6455:
6444:
6439:
6438:
6431:
6421:
6419:
6374:
6369:
6368:
6364:
6354:
6352:
6331:
6326:
6325:
6321:
6311:
6309:
6298:
6293:
6292:
6288:
6278:
6276:
6247:
6242:
6241:
6237:
6196:
6195:
6191:
6181:
6179:
6150:
6145:
6144:
6140:
6118:
6117:
6113:
6103:
6101:
6072:
6067:
6066:
6062:
6046:
6041:
6040:
6036:
6006:
6005:
6001:
5957:
5952:
5951:
5942:
5932:
5930:
5928:Quanta Magazine
5921:
5920:
5916:
5911:Wayback Machine
5901:
5897:
5887:
5885:
5878:
5874:
5873:
5869:
5854:
5853:
5849:
5830:
5829:
5828:
5824:
5815:
5813:
5808:
5807:
5803:
5780:
5779:
5775:
5758:
5751:
5749:
5739:
5738:
5731:
5724:
5709:
5708:
5704:
5697:
5682:
5681:
5677:
5662:
5661:
5657:
5650:
5632:
5627:
5623:
5615:Roshdi Rashed,
5614:
5610:
5598:
5588:
5587:
5583:
5574:
5570:
5555:
5551:
5541:
5539:
5531:
5530:
5526:
5511:
5510:
5506:
5497:
5493:
5484:
5482:
5474:
5473:
5469:
5464:
5459:
5458:
5445:
5440:
5439:
5409:
5404:
5403:
5395:
5387:
5379:
5371:
5367:
5366:. For example,
5356:
5329:
5324:
5323:
5322:All factors of
5321:
5317:
5312:
5270:
5226:
5225:
5199:
5158:proper divisors
5152:
5145:
5143:
5136:
5134:
5127:
5125:
5118:
5116:
5105:
5103:
5092:
5090:
5088:Highly abundant
5083:
5081:
5074:
5072:
5065:
5054:
4998:
4997:
4967:
4966:
4932:
4927:
4926:
4893:
4874:
4869:
4868:
4736:
4735:
4699:
4666:
4555:
4554:
4553:For 6, we have
4508:
4503:
4502:
4460:
4400:
4392:
4387:
4376:
4375:
4339:
4325:
4324:
4309:
4300:
4294:
4289:
4280:
4208:
4207:
4197:
4188:
4166:
4142:
4126:
4110:
4097:
4088:
4042:
4016:
3996:
3973:
3972:
3944:
3928:
3915:
3905:
3895:
3887:
3886:
3854:
3839:
3817:
3801:
3785:
3771:
3770:
3737:
3718:
3710:
3699:
3698:
3660:
3653:
3652:
3633:
3624:
3583:
3555:
3532:
3521:
3520:
3466:
3459:
3458:
3429:
3422:
3421:
3390:
3383:
3382:
3374:⥠81 (mod 324).
3362:is of the form
3317:Jacques LefĂšvre
3313:
3312:
3307:
3304:
3300:
3297:
3261:
3260:
3250:
3245:
3233:
3220:
3207:
3194:
3181:
3168:
3155:
3151:
3138:
3135:
3134:
3124:
3119:
3109:
3096:
3083:
3070:
3057:
3055:
3042:
3039:
3038:
3028:
3023:
3013:
3000:
2987:
2985:
2972:
2969:
2968:
2958:
2953:
2943:
2930:
2928:
2915:
2906:
2905:
2896:
2892:
2859:
2840:
2835:
2834:
2827:
2797:
2778:
2773:
2772:
2764:
2763:with odd prime
2734:
2715:
2710:
2709:
2705:
2701:
2697:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2661:
2657:
2653:
2649:
2645:
2641:
2637:
2595:
2587:
2547:
2522:
2518:
2489:
2484:
2479:
2478:
2462:
2461:
2451:
2438:
2425:
2406:
2393:
2380:
2369:
2368:
2321:
2302:
2289:
2282:
2276:
2275:
2265:
2252:
2239:
2226:
2213:
2200:
2187:
2174:
2163:
2162:
2115:
2096:
2083:
2076:
2070:
2069:
2059:
2046:
2033:
2020:
2009:
2008:
1961:
1942:
1929:
1922:
1916:
1915:
1905:
1892:
1881:
1880:
1833:
1814:
1801:
1794:
1788:
1787:
1761:
1742:
1729:
1722:
1710:
1709:
1673:
1667:
1662:
1661:
1633:
1627:
1622:
1621:
1583:
1582:
1574:
1573:
1542:
1537:
1536:
1509:
1504:
1503:
1469:
1461:
1460:
1430:
1411:
1406:
1405:
1399:infinitely many
1390:
1386:
1382:
1368:
1364:
1331:
1312:
1307:
1306:
1261:
1242:
1237:
1236:
1201:
1196:
1195:
1165:
1146:
1141:
1140:
1102:
1097:
1096:
1092:
1088:
1084:
1057:
1052:
1051:
1047:
1020:
1015:
1014:
1003:Mersenne primes
975:
970:
969:
953:
952:
915:
902:
894:
882:
881:
844:
831:
823:
811:
810:
773:
760:
752:
740:
739:
702:
689:
681:
663:
662:
654:
621:
602:
597:
596:
560:
555:
554:
524:
505:
500:
499:
493:
492:
487:
484:
480:
477:
471:
438:
411:
406:
405:
375:
356:
351:
350:
335:
296:
295:
268:
263:
262:
240:
239:
197:
196:
189:complete number
176:ÏÎλΔÎčÎżÏ áŒÏÎčΞΌÏÏ
139:
134:
133:
94:
89:
88:
35:
28:
23:
22:
18:Perfect numbers
15:
12:
11:
5:
10062:
10060:
10052:
10051:
10046:
10041:
10036:
10031:
10021:
10020:
10014:
10013:
10011:
10010:
9999:
9996:
9995:
9992:
9991:
9989:
9988:
9982:
9979:
9978:
9972:
9965:
9964:
9961:
9960:
9958:
9957:
9952:
9946:
9943:
9942:
9936:
9929:
9928:
9925:
9924:
9922:
9921:
9919:Sorting number
9916:
9914:Pancake number
9910:
9907:
9906:
9900:
9893:
9892:
9889:
9888:
9886:
9885:
9880:
9874:
9871:
9870:
9864:
9857:
9856:
9853:
9852:
9850:
9849:
9844:
9839:
9833:
9830:
9829:
9826:Binary numbers
9824:
9817:
9816:
9813:
9812:
9809:
9808:
9806:
9805:
9799:
9797:
9793:
9792:
9790:
9789:
9784:
9779:
9774:
9769:
9764:
9759:
9753:
9751:
9747:
9746:
9744:
9743:
9738:
9733:
9728:
9723:
9717:
9715:
9707:
9706:
9704:
9703:
9698:
9693:
9688:
9683:
9678:
9673:
9667:
9665:
9658:
9657:
9655:
9654:
9653:
9652:
9641:
9639:
9636:P-adic numbers
9632:
9631:
9628:
9627:
9625:
9624:
9623:
9622:
9612:
9607:
9602:
9597:
9592:
9587:
9582:
9577:
9571:
9569:
9565:
9564:
9562:
9561:
9555:
9553:
9552:Coding-related
9549:
9548:
9546:
9545:
9540:
9534:
9532:
9528:
9527:
9525:
9524:
9519:
9514:
9509:
9503:
9501:
9492:
9491:
9490:
9489:
9487:Multiplicative
9484:
9473:
9471:
9456:
9455:
9451:Numeral system
9449:
9442:
9441:
9438:
9437:
9435:
9434:
9429:
9424:
9419:
9414:
9409:
9404:
9399:
9394:
9389:
9384:
9379:
9374:
9369:
9364:
9359:
9353:
9350:
9349:
9338:
9331:
9330:
9327:
9326:
9323:
9322:
9320:
9319:
9314:
9308:
9306:
9300:
9299:
9297:
9296:
9291:
9286:
9281:
9275:
9273:
9267:
9266:
9264:
9263:
9258:
9253:
9248:
9243:
9241:Highly totient
9238:
9232:
9230:
9224:
9223:
9221:
9220:
9215:
9209:
9207:
9201:
9200:
9198:
9197:
9192:
9187:
9182:
9177:
9172:
9167:
9162:
9157:
9152:
9147:
9142:
9137:
9132:
9127:
9122:
9117:
9112:
9107:
9102:
9097:
9095:Almost perfect
9092:
9086:
9084:
9074:
9073:
9064:
9057:
9056:
9053:
9052:
9050:
9049:
9044:
9039:
9034:
9029:
9024:
9019:
9014:
9009:
9004:
8999:
8994:
8988:
8985:
8984:
8979:
8972:
8971:
8968:
8967:
8965:
8964:
8959:
8954:
8949:
8943:
8940:
8939:
8934:
8927:
8926:
8923:
8922:
8920:
8919:
8914:
8909:
8904:
8902:Stirling first
8899:
8894:
8889:
8884:
8879:
8874:
8869:
8864:
8859:
8854:
8849:
8844:
8839:
8834:
8829:
8824:
8818:
8815:
8814:
8811:
8804:
8803:
8800:
8799:
8796:
8795:
8792:
8791:
8789:
8788:
8783:
8778:
8772:
8770:
8763:
8757:
8756:
8753:
8752:
8750:
8749:
8743:
8741:
8735:
8734:
8732:
8731:
8726:
8721:
8716:
8711:
8706:
8700:
8698:
8692:
8691:
8689:
8688:
8683:
8678:
8673:
8668:
8662:
8660:
8651:
8645:
8644:
8641:
8640:
8638:
8637:
8632:
8627:
8622:
8617:
8612:
8607:
8602:
8597:
8592:
8586:
8584:
8578:
8577:
8575:
8574:
8569:
8564:
8559:
8554:
8549:
8544:
8539:
8534:
8528:
8526:
8517:
8507:
8506:
8501:
8494:
8493:
8490:
8489:
8487:
8486:
8481:
8476:
8471:
8466:
8461:
8455:
8452:
8451:
8448:
8441:
8440:
8437:
8436:
8434:
8433:
8428:
8423:
8418:
8413:
8407:
8404:
8403:
8400:
8393:
8392:
8389:
8388:
8386:
8385:
8380:
8375:
8370:
8365:
8360:
8355:
8350:
8344:
8341:
8340:
8334:
8327:
8326:
8323:
8322:
8320:
8319:
8314:
8309:
8304:
8299:
8293:
8290:
8289:
8286:
8279:
8278:
8275:
8274:
8272:
8271:
8266:
8261:
8256:
8251:
8246:
8241:
8235:
8232:
8231:
8224:
8217:
8216:
8213:
8212:
8210:
8209:
8204:
8199:
8194:
8189:
8184:
8179:
8174:
8169:
8164:
8159:
8154:
8149:
8144:
8138:
8135:
8134:
8128:
8121:
8120:
8114:
8112:
8111:
8104:
8097:
8089:
8080:
8079:
8077:
8076:
8071:
8066:
8061:
8056:
8051:
8046:
8041:
8036:
8030:
8028:
8024:
8023:
8021:
8020:
8015:
8010:
8005:
8000:
7995:
7989:
7987:
7980:
7979:
7977:
7976:
7971:
7966:
7956:
7950:
7948:
7941:
7940:
7938:
7937:
7932:
7927:
7922:
7917:
7912:
7907:
7902:
7896:
7894:
7890:
7889:
7887:
7886:
7881:
7876:
7871:
7866:
7861:
7856:
7851:
7846:
7841:
7839:Almost perfect
7836:
7830:
7828:
7824:
7823:
7821:
7820:
7815:
7810:
7805:
7800:
7795:
7790:
7785:
7780:
7775:
7770:
7765:
7760:
7754:
7752:
7748:
7747:
7740:
7738:
7736:
7735:
7730:
7725:
7720:
7715:
7710:
7704:
7702:
7698:
7697:
7694:
7692:
7691:
7684:
7677:
7669:
7663:
7662:
7634:
7628:
7622:
7619:OddPerfect.org
7616:
7608:
7589:
7584:
7578:
7558:
7557:External links
7555:
7554:
7553:
7539:
7518:
7511:
7504:
7467:
7462:
7459:
7457:
7456:
7443:
7414:
7400:
7398:
7395:
7392:
7391:
7383:
7363:
7352:Dickson, L. E.
7343:
7330:
7311:(4): 390â392.
7295:
7276:(6): 442â443.
7260:
7217:
7210:
7190:
7183:10.4171/EM/149
7176:(3): 121â126.
7156:
7137:
7124:
7071:
7024:
7021:
7017:
7011:
7007:
6987:
6938:
6891:
6856:(1): 129â146.
6835:
6806:(2): 280â286.
6786:
6727:
6676:
6657:
6638:
6607:
6564:
6529:
6498:
6485:(3): 353â359.
6465:
6429:
6362:
6319:
6286:
6235:
6189:
6138:
6111:
6060:
6034:
5999:
5940:
5914:
5903:Oddperfect.org
5895:
5867:
5856:Dickson, L. E.
5847:
5822:
5812:. Mersenne.org
5801:
5773:
5729:
5722:
5702:
5695:
5675:
5664:Dickson, L. E.
5655:
5648:
5621:
5608:
5581:
5568:
5549:
5524:
5513:Dickson, L. E.
5504:
5491:
5466:
5465:
5463:
5460:
5457:
5456:
5427:
5424:
5421:
5416:
5412:
5344:
5341:
5336:
5332:
5314:
5313:
5311:
5308:
5307:
5306:
5301:
5296:
5291:
5286:
5281:
5279:Leinster group
5276:
5269:
5266:
5235:
5144:
5135:
5126:
5117:
5104:
5091:
5082:
5073:
5064:
5053:
5050:
5049:
5048:
5041:
5034:
5018:
5013:
5008:
5005:
4979:
4974:
4959:
4947:
4944:
4939:
4935:
4911:
4908:
4905:
4900:
4896:
4892:
4887:
4884:
4881:
4877:
4857:
4856:
4855:
4841:
4840:
4839:
4827:
4824:
4821:
4817:
4813:
4810:
4807:
4803:
4799:
4796:
4793:
4789:
4785:
4782:
4779:
4775:
4771:
4768:
4765:
4761:
4757:
4754:
4751:
4747:
4743:
4732:
4720:
4717:
4712:
4708:
4705:
4702:
4696:
4691:
4687:
4684:
4681:
4678:
4675:
4672:
4669:
4663:
4658:
4655:
4650:
4645:
4642:
4637:
4632:
4629:
4624:
4619:
4616:
4611:
4606:
4603:
4598:
4593:
4590:
4585:
4580:
4577:
4572:
4567:
4564:
4535:
4532:
4529:
4526:
4523:
4520:
4515:
4511:
4491:
4484:
4459:
4456:
4443:
4442:
4441:
4440:
4424:
4421:
4418:
4415:
4412:
4407:
4403:
4399:
4395:
4390:
4386:
4383:
4363:
4360:
4357:
4354:
4351:
4346:
4342:
4338:
4335:
4332:
4322:
4305:
4298:
4291:
4285:
4278:
4272:
4258:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4231:
4227:
4224:
4221:
4218:
4215:
4193:
4186:
4180:
4162:
4156:
4138:
4132:
4122:
4116:
4106:
4093:
4086:
4080:
4079:
4078:
4077:
4065:
4062:
4059:
4056:
4049:
4045:
4041:
4036:
4033:
4030:
4023:
4019:
4015:
4010:
4003:
3999:
3995:
3990:
3985:
3982:
3970:
3955:
3952:
3947:
3943:
3940:
3935:
3931:
3927:
3922:
3918:
3912:
3908:
3902:
3898:
3894:
3884:
3870:
3866:
3863:
3860:
3857:
3851:
3846:
3842:
3838:
3835:
3832:
3829:
3824:
3820:
3816:
3813:
3808:
3804:
3800:
3797:
3792:
3788:
3784:
3781:
3778:
3768:
3755:
3750:
3747:
3744:
3740:
3736:
3731:
3728:
3725:
3721:
3717:
3713:
3709:
3706:
3696:
3689:
3678:
3673:
3669:
3666:
3663:
3645:
3635:
3629:
3622:
3610:
3609:
3608:
3597:
3590:
3586:
3582:
3577:
3573:
3569:
3562:
3558:
3554:
3549:
3545:
3539:
3535:
3531:
3528:
3515:
3514:
3513:is of the form
3508:
3494:
3483:
3477:
3472:
3469:
3455:
3440:
3435:
3432:
3418:
3407:
3401:
3396:
3393:
3375:
3357:
3351:
3322:Carl Pomerance
3308:
3305:
3299:
3296:
3293:
3257:
3253:
3249:
3246:
3240:
3236:
3232:
3227:
3223:
3219:
3214:
3210:
3206:
3201:
3197:
3193:
3188:
3184:
3180:
3175:
3171:
3167:
3162:
3158:
3152:
3150:
3145:
3141:
3137:
3136:
3131:
3127:
3123:
3120:
3116:
3112:
3108:
3103:
3099:
3095:
3090:
3086:
3082:
3077:
3073:
3069:
3064:
3060:
3056:
3054:
3049:
3045:
3041:
3040:
3035:
3031:
3027:
3024:
3020:
3016:
3012:
3007:
3003:
2999:
2994:
2990:
2986:
2984:
2979:
2975:
2971:
2970:
2965:
2961:
2957:
2954:
2950:
2946:
2942:
2937:
2933:
2929:
2927:
2922:
2918:
2914:
2913:
2880:
2877:
2874:
2871:
2866:
2862:
2858:
2853:
2850:
2847:
2843:
2815:
2812:
2809:
2804:
2800:
2796:
2791:
2788:
2785:
2781:
2770:
2752:
2749:
2746:
2741:
2737:
2733:
2728:
2725:
2722:
2718:
2690:= 3727815, ...
2687:
2679:
2671:
2663:
2655:
2647:
2639:
2621:
2617:
2613:
2610:
2607:
2602:
2598:
2594:
2590:
2586:
2583:
2580:
2577:
2574:
2569:
2565:
2562:
2559:
2554:
2550:
2546:
2543:
2540:
2537:
2534:
2529:
2525:
2521:
2515:
2512:
2509:
2504:
2501:
2496:
2492:
2487:
2458:
2454:
2450:
2445:
2441:
2437:
2432:
2428:
2424:
2421:
2418:
2413:
2409:
2405:
2400:
2396:
2392:
2387:
2383:
2379:
2376:
2374:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2323:
2320:
2317:
2314:
2309:
2305:
2301:
2296:
2292:
2288:
2285:
2283:
2281:
2278:
2277:
2272:
2268:
2264:
2259:
2255:
2251:
2246:
2242:
2238:
2233:
2229:
2225:
2220:
2216:
2212:
2207:
2203:
2199:
2194:
2190:
2186:
2181:
2177:
2173:
2170:
2168:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2117:
2114:
2111:
2108:
2103:
2099:
2095:
2090:
2086:
2082:
2079:
2077:
2075:
2072:
2071:
2066:
2062:
2058:
2053:
2049:
2045:
2040:
2036:
2032:
2027:
2023:
2019:
2016:
2014:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1963:
1960:
1957:
1954:
1949:
1945:
1941:
1936:
1932:
1928:
1925:
1923:
1921:
1918:
1917:
1912:
1908:
1904:
1899:
1895:
1891:
1888:
1886:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1835:
1832:
1829:
1826:
1821:
1817:
1813:
1808:
1804:
1800:
1797:
1795:
1793:
1790:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1763:
1760:
1757:
1754:
1749:
1745:
1741:
1736:
1732:
1728:
1725:
1723:
1721:
1718:
1717:
1695:
1692:
1686:
1682:
1679:
1676:
1670:
1646:
1642:
1639:
1636:
1630:
1602:
1598:
1595:
1590:
1586:
1555:
1552:
1549:
1545:
1524:
1521:
1516:
1512:
1487:
1484:
1481:
1476:
1472:
1468:
1448:
1445:
1442:
1437:
1433:
1429:
1424:
1421:
1418:
1414:
1379:
1378:
1349:
1346:
1343:
1338:
1334:
1330:
1325:
1322:
1319:
1315:
1279:
1276:
1273:
1268:
1264:
1260:
1255:
1252:
1249:
1245:
1233:Leonhard Euler
1229:Ibn al-Haytham
1216:
1213:
1208:
1204:
1183:
1180:
1177:
1172:
1168:
1164:
1159:
1156:
1153:
1149:
1138:
1117:
1114:
1109:
1105:
1072:
1069:
1064:
1060:
1035:
1032:
1027:
1023:
1009:, who studied
1007:Marin Mersenne
990:
987:
982:
978:
951:
948:
945:
942:
939:
936:
933:
930:
927:
922:
918:
914:
909:
905:
900:
897:
895:
893:
890:
887:
884:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
851:
847:
843:
838:
834:
829:
826:
824:
822:
819:
816:
813:
812:
809:
806:
803:
800:
797:
794:
791:
788:
785:
780:
776:
772:
767:
763:
758:
755:
753:
751:
748:
745:
742:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
709:
705:
701:
696:
692:
687:
684:
682:
680:
677:
674:
671:
670:
661:, as follows:
642:
639:
636:
633:
628:
624:
620:
615:
612:
609:
605:
575:
572:
567:
563:
542:
539:
536:
531:
527:
523:
518:
515:
512:
508:
488:
485:
479:
470:
467:
463:Pietro Cataldi
426:
423:
418:
414:
393:
390:
387:
382:
378:
374:
369:
366:
363:
359:
348:
334:
331:
320:Leonhard Euler
316:Mersenne prime
303:
283:
280:
275:
271:
247:
227:
223:
219:
216:
213:
210:
207:
204:
146:
142:
121:
118:
115:
112:
109:
106:
101:
97:
51:perfect number
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
10061:
10050:
10047:
10045:
10042:
10040:
10037:
10035:
10032:
10030:
10027:
10026:
10024:
10009:
10005:
10001:
10000:
9997:
9987:
9984:
9983:
9980:
9975:
9970:
9966:
9956:
9953:
9951:
9948:
9947:
9944:
9939:
9934:
9930:
9920:
9917:
9915:
9912:
9911:
9908:
9903:
9898:
9894:
9884:
9881:
9879:
9876:
9875:
9872:
9868:
9862:
9858:
9848:
9845:
9843:
9840:
9838:
9835:
9834:
9831:
9827:
9822:
9818:
9804:
9801:
9800:
9798:
9794:
9788:
9785:
9783:
9780:
9778:
9777:Polydivisible
9775:
9773:
9770:
9768:
9765:
9763:
9760:
9758:
9755:
9754:
9752:
9748:
9742:
9739:
9737:
9734:
9732:
9729:
9727:
9724:
9722:
9719:
9718:
9716:
9713:
9708:
9702:
9699:
9697:
9694:
9692:
9689:
9687:
9684:
9682:
9679:
9677:
9674:
9672:
9669:
9668:
9666:
9663:
9659:
9651:
9648:
9647:
9646:
9643:
9642:
9640:
9637:
9633:
9621:
9618:
9617:
9616:
9613:
9611:
9608:
9606:
9603:
9601:
9598:
9596:
9593:
9591:
9588:
9586:
9583:
9581:
9578:
9576:
9573:
9572:
9570:
9566:
9560:
9557:
9556:
9554:
9550:
9544:
9541:
9539:
9536:
9535:
9533:
9531:Digit product
9529:
9523:
9520:
9518:
9515:
9513:
9510:
9508:
9505:
9504:
9502:
9500:
9496:
9488:
9485:
9483:
9480:
9479:
9478:
9475:
9474:
9472:
9470:
9465:
9461:
9457:
9452:
9447:
9443:
9433:
9430:
9428:
9425:
9423:
9420:
9418:
9415:
9413:
9410:
9408:
9405:
9403:
9400:
9398:
9395:
9393:
9390:
9388:
9385:
9383:
9380:
9378:
9375:
9373:
9370:
9368:
9367:ErdĆsâNicolas
9365:
9363:
9360:
9358:
9355:
9354:
9351:
9346:
9342:
9336:
9332:
9318:
9315:
9313:
9310:
9309:
9307:
9305:
9301:
9295:
9292:
9290:
9287:
9285:
9282:
9280:
9277:
9276:
9274:
9272:
9268:
9262:
9259:
9257:
9254:
9252:
9249:
9247:
9244:
9242:
9239:
9237:
9234:
9233:
9231:
9229:
9225:
9219:
9216:
9214:
9211:
9210:
9208:
9206:
9202:
9196:
9193:
9191:
9188:
9186:
9185:Superabundant
9183:
9181:
9178:
9176:
9173:
9171:
9168:
9166:
9163:
9161:
9158:
9156:
9153:
9151:
9148:
9146:
9143:
9141:
9138:
9136:
9133:
9131:
9128:
9126:
9123:
9121:
9118:
9116:
9113:
9111:
9108:
9106:
9103:
9101:
9098:
9096:
9093:
9091:
9088:
9087:
9085:
9083:
9079:
9075:
9071:
9067:
9062:
9058:
9048:
9045:
9043:
9040:
9038:
9035:
9033:
9030:
9028:
9025:
9023:
9020:
9018:
9015:
9013:
9010:
9008:
9005:
9003:
9000:
8998:
8995:
8993:
8990:
8989:
8986:
8982:
8977:
8973:
8963:
8960:
8958:
8955:
8953:
8950:
8948:
8945:
8944:
8941:
8937:
8932:
8928:
8918:
8915:
8913:
8910:
8908:
8905:
8903:
8900:
8898:
8895:
8893:
8890:
8888:
8885:
8883:
8880:
8878:
8875:
8873:
8870:
8868:
8865:
8863:
8860:
8858:
8855:
8853:
8850:
8848:
8845:
8843:
8840:
8838:
8835:
8833:
8830:
8828:
8825:
8823:
8820:
8819:
8816:
8809:
8805:
8787:
8784:
8782:
8779:
8777:
8774:
8773:
8771:
8767:
8764:
8762:
8761:4-dimensional
8758:
8748:
8745:
8744:
8742:
8740:
8736:
8730:
8727:
8725:
8722:
8720:
8717:
8715:
8712:
8710:
8707:
8705:
8702:
8701:
8699:
8697:
8693:
8687:
8684:
8682:
8679:
8677:
8674:
8672:
8671:Centered cube
8669:
8667:
8664:
8663:
8661:
8659:
8655:
8652:
8650:
8649:3-dimensional
8646:
8636:
8633:
8631:
8628:
8626:
8623:
8621:
8618:
8616:
8613:
8611:
8608:
8606:
8603:
8601:
8598:
8596:
8593:
8591:
8588:
8587:
8585:
8583:
8579:
8573:
8570:
8568:
8565:
8563:
8560:
8558:
8555:
8553:
8550:
8548:
8545:
8543:
8540:
8538:
8535:
8533:
8530:
8529:
8527:
8525:
8521:
8518:
8516:
8515:2-dimensional
8512:
8508:
8504:
8499:
8495:
8485:
8482:
8480:
8477:
8475:
8472:
8470:
8467:
8465:
8462:
8460:
8459:Nonhypotenuse
8457:
8456:
8453:
8446:
8442:
8432:
8429:
8427:
8424:
8422:
8419:
8417:
8414:
8412:
8409:
8408:
8405:
8398:
8394:
8384:
8381:
8379:
8376:
8374:
8371:
8369:
8366:
8364:
8361:
8359:
8356:
8354:
8351:
8349:
8346:
8345:
8342:
8337:
8332:
8328:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8300:
8298:
8295:
8294:
8291:
8284:
8280:
8270:
8267:
8265:
8262:
8260:
8257:
8255:
8252:
8250:
8247:
8245:
8242:
8240:
8237:
8236:
8233:
8228:
8222:
8218:
8208:
8205:
8203:
8200:
8198:
8197:Perfect power
8195:
8193:
8190:
8188:
8187:Seventh power
8185:
8183:
8180:
8178:
8175:
8173:
8170:
8168:
8165:
8163:
8160:
8158:
8155:
8153:
8150:
8148:
8145:
8143:
8140:
8139:
8136:
8131:
8126:
8122:
8118:
8110:
8105:
8103:
8098:
8096:
8091:
8090:
8087:
8075:
8072:
8070:
8067:
8065:
8062:
8060:
8057:
8055:
8052:
8050:
8047:
8045:
8042:
8040:
8037:
8035:
8032:
8031:
8029:
8025:
8019:
8016:
8014:
8013:Polydivisible
8011:
8009:
8006:
8004:
8001:
7999:
7996:
7994:
7991:
7990:
7988:
7985:
7981:
7975:
7972:
7970:
7967:
7964:
7960:
7957:
7955:
7952:
7951:
7949:
7946:
7942:
7936:
7933:
7931:
7928:
7926:
7923:
7921:
7918:
7916:
7915:Superabundant
7913:
7911:
7908:
7906:
7903:
7901:
7898:
7897:
7895:
7891:
7885:
7884:ErdĆsâNicolas
7882:
7880:
7877:
7875:
7872:
7870:
7867:
7865:
7862:
7860:
7857:
7855:
7852:
7850:
7847:
7845:
7842:
7840:
7837:
7835:
7832:
7831:
7829:
7825:
7819:
7816:
7814:
7811:
7809:
7806:
7804:
7801:
7799:
7796:
7794:
7793:Perfect power
7791:
7789:
7786:
7784:
7781:
7779:
7776:
7774:
7771:
7769:
7766:
7764:
7761:
7759:
7756:
7755:
7753:
7749:
7744:
7734:
7731:
7729:
7726:
7724:
7721:
7719:
7716:
7714:
7711:
7709:
7706:
7705:
7703:
7699:
7690:
7685:
7683:
7678:
7676:
7671:
7670:
7667:
7653:on 2013-05-31
7652:
7648:
7644:
7640:
7635:
7632:
7629:
7626:
7623:
7620:
7617:
7615:
7609:
7604:
7603:
7598:
7595:
7590:
7588:
7585:
7583:
7580:David Moews:
7579:
7575:
7571:
7570:
7565:
7561:
7560:
7556:
7550:
7546:
7542:
7540:1-4020-2546-7
7536:
7532:
7527:
7526:
7519:
7516:
7512:
7509:
7505:
7501:
7497:
7492:
7487:
7483:
7479:
7478:
7473:
7468:
7465:
7464:
7460:
7453:
7449:
7444:
7440:
7436:
7432:
7428:
7424:
7420:
7415:
7412:
7408:
7407:
7402:
7401:
7396:
7386:
7384:9780824796969
7380:
7376:
7375:
7367:
7364:
7359:
7358:
7353:
7347:
7344:
7340:
7337:H. Novarese.
7334:
7331:
7326:
7322:
7318:
7314:
7310:
7306:
7299:
7296:
7291:
7287:
7283:
7279:
7275:
7271:
7264:
7261:
7256:
7252:
7248:
7244:
7240:
7236:
7232:
7228:
7221:
7218:
7213:
7211:9781118188613
7207:
7203:
7202:
7194:
7191:
7184:
7179:
7175:
7171:
7167:
7160:
7157:
7152:
7148:
7141:
7138:
7134:
7128:
7125:
7120:
7116:
7112:
7108:
7104:
7100:
7095:
7090:
7086:
7082:
7075:
7072:
7067:
7063:
7059:
7055:
7051:
7047:
7046:
7038:
7022:
7019:
7015:
7009:
7005:
6991:
6988:
6983:
6979:
6975:
6971:
6966:
6961:
6957:
6953:
6949:
6942:
6939:
6934:
6930:
6926:
6922:
6918:
6914:
6913:
6905:
6898:
6896:
6892:
6887:
6883:
6879:
6875:
6871:
6867:
6863:
6859:
6855:
6851:
6850:
6845:
6839:
6836:
6831:
6827:
6823:
6819:
6814:
6809:
6805:
6801:
6797:
6790:
6787:
6782:
6778:
6774:
6770:
6765:
6760:
6756:
6752:
6751:
6743:
6736:
6734:
6732:
6728:
6723:
6719:
6715:
6711:
6707:
6703:
6699:
6695:
6691:
6687:
6680:
6677:
6672:
6668:
6661:
6658:
6654:(6): 523-527.
6653:
6649:
6642:
6639:
6626:
6622:
6618:
6611:
6608:
6595:
6590:
6586:
6582:
6575:
6568:
6565:
6559:
6554:
6550:
6546:
6545:
6540:
6533:
6530:
6517:
6513:
6509:
6502:
6499:
6493:
6488:
6484:
6480:
6476:
6469:
6466:
6454:
6450:
6443:
6436:
6434:
6430:
6418:
6414:
6410:
6406:
6402:
6398:
6393:
6388:
6384:
6380:
6373:
6366:
6363:
6350:
6345:
6341:
6337:
6330:
6323:
6320:
6308:
6304:
6297:
6290:
6287:
6274:
6269:
6265:
6261:
6257:
6253:
6246:
6239:
6236:
6230:
6226:
6222:
6218:
6213:
6208:
6204:
6200:
6193:
6190:
6177:
6172:
6168:
6164:
6160:
6156:
6149:
6142:
6139:
6134:
6130:
6126:
6122:
6115:
6112:
6099:
6094:
6090:
6086:
6082:
6078:
6071:
6064:
6061:
6056:
6052:
6045:
6038:
6035:
6030:
6026:
6022:
6018:
6014:
6011:(in German).
6010:
6003:
6000:
5995:
5991:
5987:
5983:
5978:
5973:
5969:
5965:
5964:
5956:
5949:
5947:
5945:
5941:
5929:
5925:
5918:
5915:
5912:
5908:
5904:
5899:
5896:
5884:
5877:
5871:
5868:
5863:
5862:
5857:
5851:
5848:
5842:
5841:
5836:
5833:
5826:
5823:
5811:
5805:
5802:
5798:
5794:
5793:
5788:
5784:
5777:
5774:
5769:
5763:
5748:
5747:
5742:
5736:
5734:
5730:
5725:
5723:88-8358-537-2
5719:
5715:
5714:
5706:
5703:
5698:
5696:0-19-515799-0
5692:
5688:
5687:
5679:
5676:
5671:
5670:
5665:
5659:
5656:
5651:
5649:0-486-20430-8
5645:
5641:
5640:
5635:
5630:
5625:
5622:
5618:
5612:
5609:
5604:
5597:
5596:
5593:
5585:
5582:
5578:
5572:
5569:
5565:
5561:
5560:
5553:
5550:
5538:
5534:
5528:
5525:
5520:
5519:
5514:
5508:
5505:
5501:
5495:
5492:
5481:
5477:
5471:
5468:
5461:
5452:
5448:
5425:
5422:
5419:
5414:
5410:
5399:
5391:
5383:
5377:
5364:
5360:
5342:
5339:
5334:
5330:
5319:
5316:
5309:
5305:
5302:
5300:
5297:
5295:
5292:
5290:
5287:
5285:
5282:
5280:
5277:
5275:
5272:
5271:
5267:
5265:
5263:
5262:weird numbers
5258:
5253:
5251:
5223:
5218:
5214:
5210:
5206:
5202:
5198:
5194:
5189:
5187:
5183:
5179:
5175:
5171:
5167:
5163:
5159:
5151:
5142:
5133:
5124:
5115:
5111:
5102:
5098:
5097:Superabundant
5089:
5080:
5071:
5062:
5061:Euler diagram
5058:
5051:
5046:
5042:
5039:
5035:
5032:
5011:
5003:
4995:
4977:
4972:
4965:is less than
4964:
4960:
4945:
4942:
4937:
4933:
4925:
4906:
4903:
4898:
4894:
4885:
4882:
4879:
4875:
4866:
4862:
4858:
4853:
4849:
4848:
4846:
4842:
4825:
4822:
4819:
4815:
4811:
4808:
4805:
4801:
4797:
4794:
4791:
4787:
4783:
4780:
4777:
4773:
4769:
4766:
4763:
4759:
4755:
4752:
4749:
4745:
4741:
4733:
4718:
4715:
4710:
4706:
4703:
4700:
4694:
4689:
4685:
4682:
4679:
4676:
4673:
4670:
4667:
4661:
4656:
4653:
4648:
4643:
4640:
4635:
4630:
4627:
4622:
4617:
4614:
4609:
4604:
4601:
4596:
4591:
4588:
4583:
4578:
4575:
4570:
4565:
4562:
4552:
4551:
4549:
4533:
4530:
4527:
4521:
4513:
4509:
4500:
4496:
4492:
4489:
4488:Gallardo 2010
4485:
4482:
4481:Makowski 1962
4478:
4474:
4473:
4472:
4470:
4466:
4458:Minor results
4457:
4454:
4450:
4448:
4422:
4419:
4416:
4413:
4410:
4405:
4401:
4397:
4393:
4388:
4384:
4381:
4361:
4358:
4355:
4352:
4349:
4344:
4340:
4336:
4333:
4330:
4323:
4320:
4317:
4316:
4313:
4308:
4304:
4297:
4292:
4288:
4284:
4277:
4273:
4256:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4229:
4222:
4219:
4216:
4205:
4201:
4196:
4192:
4185:
4181:
4178:
4174:
4170:
4165:
4161:
4157:
4154:
4150:
4146:
4141:
4137:
4133:
4130:
4125:
4121:
4117:
4114:
4109:
4105:
4101:
4100:
4099:
4096:
4092:
4085:
4063:
4060:
4057:
4054:
4047:
4043:
4039:
4034:
4031:
4028:
4021:
4017:
4013:
4008:
4001:
3997:
3993:
3988:
3983:
3980:
3971:
3953:
3950:
3945:
3941:
3938:
3933:
3929:
3925:
3920:
3916:
3910:
3906:
3900:
3896:
3892:
3885:
3868:
3864:
3861:
3858:
3855:
3849:
3844:
3840:
3836:
3833:
3830:
3827:
3822:
3818:
3814:
3811:
3806:
3802:
3798:
3795:
3790:
3786:
3782:
3779:
3776:
3769:
3748:
3745:
3742:
3738:
3734:
3729:
3726:
3723:
3719:
3711:
3707:
3704:
3697:
3694:
3690:
3676:
3671:
3667:
3664:
3661:
3650:
3646:
3643:
3639:
3636:
3632:
3628:
3621:
3617:
3614:
3613:
3611:
3595:
3588:
3584:
3580:
3575:
3571:
3567:
3560:
3556:
3552:
3547:
3543:
3537:
3533:
3529:
3526:
3519:
3518:
3517:
3516:
3512:
3509:
3506:
3502:
3498:
3495:
3481:
3475:
3470:
3467:
3456:
3438:
3433:
3430:
3419:
3405:
3399:
3394:
3391:
3380:
3376:
3373:
3369:
3365:
3361:
3358:
3355:
3352:
3349:
3346:
3345:
3344:
3342:
3337:
3335:
3331:
3327:
3323:
3318:
3311:
3294:
3292:
3290:
3286:
3281:
3279:
3274:
3255:
3252:1111111000000
3251:
3247:
3238:
3234:
3230:
3225:
3221:
3217:
3212:
3208:
3204:
3199:
3195:
3191:
3186:
3182:
3178:
3173:
3169:
3165:
3160:
3156:
3148:
3143:
3139:
3129:
3125:
3121:
3114:
3110:
3106:
3101:
3097:
3093:
3088:
3084:
3080:
3075:
3071:
3067:
3062:
3058:
3052:
3047:
3043:
3033:
3029:
3025:
3018:
3014:
3010:
3005:
3001:
2997:
2992:
2988:
2982:
2977:
2973:
2963:
2959:
2955:
2948:
2944:
2940:
2935:
2931:
2925:
2920:
2916:
2903:
2899:
2878:
2872:
2869:
2864:
2860:
2851:
2848:
2845:
2841:
2831:
2810:
2807:
2802:
2798:
2789:
2786:
2783:
2779:
2768:
2747:
2744:
2739:
2735:
2726:
2723:
2720:
2716:
2695:
2634:
2619:
2615:
2608:
2605:
2600:
2596:
2588:
2584:
2581:
2578:
2575:
2572:
2567:
2560:
2557:
2552:
2548:
2541:
2535:
2532:
2527:
2523:
2513:
2510:
2507:
2502:
2499:
2494:
2490:
2485:
2475:
2456:
2452:
2448:
2443:
2439:
2435:
2430:
2426:
2422:
2419:
2416:
2411:
2407:
2403:
2398:
2394:
2390:
2385:
2381:
2377:
2375:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2324:
2315:
2312:
2307:
2303:
2294:
2290:
2286:
2284:
2279:
2270:
2266:
2262:
2257:
2253:
2249:
2244:
2240:
2236:
2231:
2227:
2223:
2218:
2214:
2210:
2205:
2201:
2197:
2192:
2188:
2184:
2179:
2175:
2171:
2169:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2118:
2109:
2106:
2101:
2097:
2088:
2084:
2080:
2078:
2073:
2064:
2060:
2056:
2051:
2047:
2043:
2038:
2034:
2030:
2025:
2021:
2017:
2015:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1964:
1955:
1952:
1947:
1943:
1934:
1930:
1926:
1924:
1919:
1910:
1906:
1902:
1897:
1893:
1889:
1887:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1836:
1827:
1824:
1819:
1815:
1806:
1802:
1798:
1796:
1791:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1764:
1755:
1752:
1747:
1743:
1734:
1730:
1726:
1724:
1719:
1707:
1693:
1690:
1684:
1680:
1677:
1674:
1668:
1644:
1640:
1637:
1634:
1628:
1619:
1600:
1596:
1593:
1588:
1584:
1571:
1553:
1550:
1547:
1543:
1522:
1519:
1514:
1510:
1501:
1482:
1479:
1474:
1470:
1443:
1440:
1435:
1431:
1422:
1419:
1416:
1412:
1402:
1400:
1396:
1376:
1371:
1363:
1362:
1361:
1344:
1341:
1336:
1332:
1323:
1320:
1317:
1313:
1304:
1299:
1297:
1293:
1274:
1271:
1266:
1262:
1253:
1250:
1247:
1243:
1234:
1230:
1214:
1211:
1206:
1202:
1178:
1175:
1170:
1166:
1157:
1154:
1151:
1147:
1136:
1134:
1129:
1115:
1112:
1107:
1103:
1083:with a prime
1070:
1067:
1062:
1058:
1033:
1030:
1025:
1021:
1012:
1011:number theory
1008:
1004:
1001:are known as
988:
985:
980:
976:
966:
949:
946:
943:
940:
937:
934:
928:
925:
920:
916:
907:
903:
898:
896:
891:
888:
885:
878:
875:
872:
869:
866:
863:
857:
854:
849:
845:
836:
832:
827:
825:
820:
817:
814:
807:
804:
801:
798:
795:
792:
786:
783:
778:
774:
765:
761:
756:
754:
749:
746:
743:
736:
733:
730:
727:
724:
721:
715:
712:
707:
703:
694:
690:
685:
683:
678:
675:
672:
660:
640:
634:
631:
626:
622:
613:
610:
607:
603:
593:
591:
590:
573:
570:
565:
561:
537:
534:
529:
525:
516:
513:
510:
506:
497:
491:
476:
468:
466:
464:
460:
456:
452:
448:
444:
424:
421:
416:
412:
388:
385:
380:
376:
367:
364:
361:
357:
346:
344:
340:
332:
330:
327:
325:
321:
317:
301:
281:
278:
273:
269:
261:
245:
225:
221:
214:
211:
208:
202:
194:
190:
186:
182:
172:
171:
164:
162:
144:
140:
119:
116:
113:
107:
99:
95:
86:
81:
79:
75:
71:
67:
62:
60:
56:
52:
48:
47:number theory
39:
33:
19:
9741:Transposable
9605:Narcissistic
9512:Digital root
9432:Super-Poulet
9392:JordanâPĂłlya
9341:prime factor
9283:
9246:Noncototient
9213:Almost prime
9195:Superperfect
9170:Refactorable
9165:Quasiperfect
9149:
9140:Hyperperfect
8981:Pseudoprimes
8952:WallâSunâSun
8887:Ordered Bell
8857:FussâCatalan
8769:non-centered
8719:Dodecahedral
8696:non-centered
8582:non-centered
8484:Wolstenholme
8229:× 2 ± 1
8226:
8225:Of the form
8192:Eighth power
8172:Fourth power
8074:Superperfect
8069:Refactorable
7864:Superperfect
7859:Hyperperfect
7844:Quasiperfect
7833:
7728:Prime factor
7655:. Retrieved
7651:the original
7642:
7600:
7567:
7524:
7514:
7507:
7481:
7475:
7451:
7447:
7422:
7418:
7404:
7373:
7366:
7356:
7346:
7338:
7333:
7308:
7304:
7298:
7273:
7269:
7263:
7230:
7226:
7220:
7200:
7193:
7173:
7169:
7159:
7150:
7146:
7140:
7132:
7127:
7087:(1): 15â21.
7084:
7080:
7074:
7052:(1): 25â28.
7049:
7043:
6990:
6958:(1): 13â15.
6955:
6951:
6941:
6919:(1): 70â76.
6916:
6910:
6853:
6847:
6838:
6803:
6799:
6789:
6754:
6748:
6692:(1): 52â53.
6689:
6685:
6679:
6670:
6666:
6660:
6651:
6647:
6641:
6629:. Retrieved
6624:
6620:
6610:
6598:. Retrieved
6584:
6580:
6567:
6548:
6542:
6532:
6520:. Retrieved
6515:
6511:
6501:
6482:
6478:
6468:
6456:. Retrieved
6452:
6448:
6420:. Retrieved
6392:math/0602485
6382:
6378:
6365:
6353:. Retrieved
6339:
6335:
6322:
6310:. Retrieved
6306:
6302:
6289:
6277:. Retrieved
6255:
6251:
6238:
6202:
6198:
6192:
6180:. Retrieved
6158:
6154:
6141:
6124:
6120:
6114:
6102:. Retrieved
6080:
6076:
6063:
6054:
6050:
6037:
6012:
6008:
6002:
5967:
5961:
5933:10 September
5931:. Retrieved
5927:
5917:
5898:
5886:. Retrieved
5882:
5870:
5860:
5850:
5838:
5825:
5814:. Retrieved
5810:"GIMPS Home"
5804:
5790:
5776:
5750:. Retrieved
5744:
5712:
5705:
5685:
5678:
5668:
5658:
5638:
5624:
5616:
5611:
5602:
5595:
5591:
5584:
5571:
5557:
5552:
5540:. Retrieved
5536:
5527:
5517:
5507:
5494:
5483:. Retrieved
5479:
5470:
5397:
5389:
5381:
5362:
5318:
5254:
5216:
5212:
5208:
5204:
5200:
5190:
5169:
5155:
5131:
5038:digital root
4993:
4962:
4924:Fermat prime
4844:
4547:
4498:
4476:
4461:
4452:
4444:
4318:
4311:
4306:
4302:
4295:
4286:
4282:
4275:
4203:
4199:
4194:
4190:
4183:
4176:
4172:
4168:
4163:
4159:
4152:
4139:
4135:
4123:
4119:
4107:
4103:
4094:
4090:
4083:
4081:
3692:
3648:
3637:
3630:
3626:
3619:
3615:
3510:
3504:
3500:
3496:
3378:
3371:
3367:
3363:
3359:
3353:
3347:
3340:
3338:
3314:
3282:
3275:
2904:
2897:
2832:
2694:digital root
2635:
2476:
1708:
1403:
1380:
1300:
1130:
967:
659:prime number
594:
587:
498:proved that
494:
336:
328:
188:
184:
180:
169:
165:
82:
63:
50:
44:
9762:Extravagant
9757:Equidigital
9712:permutation
9671:Palindromic
9645:Automorphic
9543:Sum-product
9522:Sum-product
9477:Persistence
9372:ErdĆsâWoods
9294:Untouchable
9175:Semiperfect
9125:Hemiperfect
8786:Tesseractic
8724:Icosahedral
8704:Tetrahedral
8635:Dodecagonal
8336:Recursively
8207:Prime power
8182:Sixth power
8177:Fifth power
8157:Power of 10
8115:Classes of
7998:Extravagant
7993:Equidigital
7954:Untouchable
7874:Semiperfect
7854:Hemiperfect
7783:Square-free
7647:Brady Haran
7643:Numberphile
7513:Riesel, H.
6600:29 November
6015:: 202â211.
5883:Harvard.edu
5193:fixed point
5156:The sum of
5045:square-free
4495:reciprocals
4465:Richard Guy
4206:twos, then
3695:exceeds 10.
3651:is at most
3644:4) (Euler).
1391:2 Ă (2 â 1)
455:City of God
260:of the form
258:is a prime
85:aliquot sum
10023:Categories
9974:Graphemics
9847:Pernicious
9701:Undulating
9676:Pandigital
9650:Trimorphic
9251:Nontotient
9100:Arithmetic
8714:Octahedral
8615:Heptagonal
8605:Pentagonal
8590:Triangular
8431:SierpiĆski
8353:Jacobsthal
8152:Power of 3
8147:Power of 2
8034:Arithmetic
8027:Other sets
7986:-dependent
7657:2013-04-02
7549:1079.11001
7270:Arch. Math
7170:Elem. Math
7147:Elem. Math
7094:1706.09341
6673:: 896â904.
6631:7 December
6594:2303.11974
6312:6 December
6212:1810.11734
5994:1263.11005
5816:2022-07-21
5485:2024-03-21
5462:References
5378:âthat is,
5220:, and the
5174:numerology
2702:1 + 9 = 10
1535:) and the
1133:Nicomachus
586:is prime (
473:See also:
343:Nicomachus
9731:Parasitic
9580:Factorion
9507:Digit sum
9499:Digit sum
9317:Fortunate
9304:Primorial
9218:Semiprime
9155:Practical
9120:Descartes
9115:Deficient
9105:Betrothed
8947:Wieferich
8776:Pentatope
8739:pyramidal
8630:Decagonal
8625:Nonagonal
8620:Octagonal
8610:Hexagonal
8469:Practical
8416:Congruent
8348:Fibonacci
8312:Loeschian
8064:Descartes
8039:Deficient
7974:Betrothed
7879:Practical
7768:Semiprime
7763:Composite
7602:MathWorld
7574:EMS Press
7439:115983363
7325:122353640
7305:Math. Ann
7290:122525522
7255:125545112
7153:(5): 109.
7119:119175632
7111:1730-6302
7058:0015-0517
7023:β
7010:α
6974:1088-6826
6925:0015-0517
6886:122452828
6870:1435-5345
6822:1446-8107
6773:0025-5718
6722:121251041
6706:1420-8938
6518:: A14âA22
6355:13 August
6057:(4): 244.
6029:120754476
5986:0025-5718
5840:MathWorld
5420:−
5340:−
5162:deficient
5150:Deficient
5141:Composite
5043:The only
4883:−
4704:⋅
4510:σ
4447:Sylvester
4445:In 1888,
4334:≤
4257:α
4243:≤
4237:≤
4220:−
4202:ones and
4147:3) or 2 (
4061:
4032:⋯
3926:⋯
3862:−
3850:≥
3831:⋯
3777:α
3735:−
3665:−
3568:⋯
3538:α
3126:111110000
2870:−
2849:−
2808:−
2787:−
2745:−
2724:−
2706:1 + 0 = 1
2606:−
2585:×
2542:×
2533:−
2500:−
2420:⋯
2348:⋯
2313:−
2142:⋯
2107:−
1988:⋯
1953:−
1825:−
1753:−
1691:−
1638:−
1551:−
1520:−
1480:−
1441:−
1420:−
1395:not known
1342:−
1321:−
1272:−
1251:−
1212:−
1176:−
1155:−
1113:−
1068:−
1031:−
986:−
941:×
926:−
870:×
855:−
799:×
784:−
728:×
713:−
632:−
611:−
571:−
535:−
514:−
449:, and by
422:−
386:−
365:−
279:−
168:Euclid's
141:σ
96:σ
9803:Friedman
9736:Primeval
9681:Repdigit
9638:-related
9585:Kaprekar
9559:Meertens
9482:Additive
9469:dynamics
9377:Friendly
9289:Sociable
9279:Amicable
9090:Abundant
9070:dynamics
8892:Schröder
8882:Narayana
8852:Eulerian
8842:Delannoy
8837:Dedekind
8658:centered
8524:centered
8411:Amenable
8368:Narayana
8358:Leonardo
8254:Mersenne
8202:Powerful
8142:Achilles
8049:Solitary
8044:Friendly
7969:Sociable
7959:Amicable
7947:-related
7900:Abundant
7798:Achilles
7788:Powerful
7701:Overview
7454:: 69â72.
7406:Elements
7403:Euclid,
7354:(1919).
6581:Integers
6522:23 March
6512:Integers
6458:7 August
6449:Integers
6422:30 March
6303:Integers
6279:30 March
6229:62885986
6182:30 March
6104:30 March
5907:Archived
5858:(1919).
5762:cite web
5666:(1919).
5636:(1925).
5515:(1919).
5480:oeis.org
5268:See also
5182:sociable
5178:amicable
5166:abundant
5070:Abundant
5029:, using
4992:, where
4449:stated:
4301:= ... =
4118:Not all
4102:Not all
3350:> 10.
2280:33550336
589:Elements
170:Elements
59:divisors
9976:related
9940:related
9904:related
9902:Sorting
9787:Vampire
9772:Harshad
9714:related
9686:Repunit
9600:Lychrel
9575:Dudeney
9427:StĂžrmer
9422:Sphenic
9407:Regular
9345:divisor
9284:Perfect
9180:Sublime
9150:Perfect
8877:Motzkin
8832:Catalan
8373:Padovan
8307:Leyland
8302:Idoneal
8297:Hilbert
8269:Woodall
8054:Sublime
8008:Harshad
7834:Perfect
7818:Unusual
7808:Regular
7778:Sphenic
7713:Divisor
7627:(GIMPS)
7576:, 2001
7500:2005530
7397:Sources
7247:3619053
7066:0354538
6982:0292740
6933:0786364
6878:0044579
6830:0869751
6781:2904601
6714:0258723
6627:: 23â30
6417:2767519
6397:Bibcode
6260:Bibcode
6163:Bibcode
6085:Bibcode
5888:16 June
5752:28 July
5451:A002515
5449::
5195:of the
5170:perfect
5148:
5139:
5132:Perfect
5130:
5121:
5108:
5095:
5086:
5077:
5068:
4315:, then
4134:If all
3618:,
3612:where:
3503:, then
1373:in the
1370:A000043
333:History
181:perfect
159:is the
9842:Odious
9767:Frugal
9721:Cyclic
9710:Digit-
9417:Smooth
9402:Pronic
9362:Cyclic
9339:Other
9312:Euclid
8962:Wilson
8936:Primes
8595:Square
8464:Polite
8426:Riesel
8421:Knödel
8383:Perrin
8264:Thabit
8249:Fermat
8239:Cullen
8162:Square
8130:Powers
8003:Frugal
7963:Triple
7803:Smooth
7773:Pronic
7547:
7537:
7498:
7437:
7381:
7323:
7288:
7253:
7245:
7208:
7117:
7109:
7064:
7056:
6980:
6972:
6931:
6923:
6884:
6876:
6868:
6828:
6820:
6779:
6771:
6720:
6712:
6704:
6415:
6227:
6027:
5992:
5984:
5720:
5693:
5646:
5146:
5137:
5128:
5119:
5106:
5093:
5084:
5075:
5066:
4838:, etc.
2704:, and
2658:= 8128
1194:where
1131:While
496:Euclid
447:Origen
404:where
193:Euclid
132:where
9883:Prime
9878:Lucky
9867:sieve
9796:Other
9782:Smith
9662:Digit
9620:Happy
9595:Keith
9568:Other
9412:Rough
9382:Giuga
8847:Euler
8709:Cubic
8363:Lucas
8259:Proth
8018:Smith
7935:Weird
7813:Rough
7758:Prime
7533:â98.
7496:JSTOR
7435:S2CID
7321:S2CID
7286:S2CID
7251:S2CID
7243:JSTOR
7115:S2CID
7089:arXiv
7040:(PDF)
6907:(PDF)
6882:S2CID
6745:(PDF)
6718:S2CID
6589:arXiv
6577:(PDF)
6445:(PDF)
6413:S2CID
6387:arXiv
6375:(PDF)
6332:(PDF)
6299:(PDF)
6248:(PDF)
6225:S2CID
6207:arXiv
6151:(PDF)
6073:(PDF)
6047:(PDF)
6025:S2CID
5958:(PDF)
5879:(PDF)
5599:(PDF)
5542:9 May
5374:is a
5310:Notes
5123:Weird
5112:and
5099:and
3287:(cf.
3030:11100
2682:= 903
2650:= 496
1303:GIMPS
950:8128.
653:with
347:every
187:, or
185:ideal
53:is a
9837:Evil
9517:Self
9467:and
9357:Blum
9068:and
8872:Lobb
8827:Cake
8822:Bell
8572:Star
8479:Ulam
8378:Pell
8167:Cube
7984:Base
7611:OEIS
7535:ISBN
7452:1937
7423:1941
7379:ISBN
7206:ISBN
7107:ISSN
7054:ISSN
6970:ISSN
6921:ISSN
6866:ISSN
6818:ISSN
6769:ISSN
6702:ISSN
6633:2018
6602:2023
6524:2021
6460:2021
6424:2011
6357:2015
6314:2021
6281:2011
6184:2011
6106:2011
5982:ISSN
5935:2020
5890:2023
5768:link
5754:2024
5718:ISBN
5691:ISBN
5644:ISBN
5544:2018
5447:OEIS
5215:) â
5207:) =
4493:The
4385:<
4374:and
4182:If (
4055:<
3939:<
3708:<
3140:8128
2688:2730
2674:= 55
2642:= 28
2366:8191
2360:8190
2354:8189
2074:8128
1616:-th
1568:-th
1498:-th
1375:OEIS
1360:for
78:8128
76:and
49:, a
9955:Ban
9343:or
8862:Lah
7545:Zbl
7486:doi
7427:doi
7313:doi
7309:131
7278:doi
7235:doi
7178:doi
7099:doi
7085:156
6960:doi
6858:doi
6854:188
6808:doi
6759:doi
6694:doi
6553:doi
6487:doi
6405:doi
6344:doi
6268:doi
6217:doi
6171:doi
6129:doi
6093:doi
6017:doi
5990:Zbl
5972:doi
5556:In
5400:+ 1
5392:+ 1
5384:+ 1
5359:mod
4550:):
4467:'s
4293:If
4149:mod
4145:mod
4131:5).
4129:mod
4115:3).
4113:mod
3865:224
3642:mod
3291:).
3044:496
2960:110
2900:â 1
2769:all
2666:= 3
2656:127
2453:127
2440:125
2427:123
2160:127
2154:126
2148:125
1920:496
1706:):
1137:all
944:127
879:496
191:).
74:496
45:In
10025::
7645:.
7641:.
7599:.
7572:,
7566:,
7543:.
7531:15
7494:.
7482:27
7480:.
7474:.
7450:.
7433:.
7421:.
7319:.
7307:.
7284:.
7272:.
7249:.
7241:.
7231:83
7229:.
7174:65
7172:.
7168:.
7151:17
7149:.
7113:.
7105:.
7097:.
7083:.
7062:MR
7060:.
7050:13
7048:.
7042:.
6978:MR
6976:.
6968:.
6956:32
6954:.
6950:.
6929:MR
6927:.
6917:23
6915:.
6909:.
6894:^
6880:.
6874:MR
6872:.
6864:.
6852:.
6826:MR
6824:.
6816:.
6804:42
6802:.
6798:.
6777:MR
6775:.
6767:.
6755:81
6753:.
6747:.
6730:^
6716:.
6710:MR
6708:.
6700:.
6690:21
6688:.
6671:14
6669:.
6652:16
6650:.
6625:16
6623:.
6619:.
6587:.
6585:23
6583:.
6579:.
6549:83
6547:.
6541:.
6514:.
6510:.
6483:89
6481:.
6477:.
6453:21
6451:.
6447:.
6432:^
6411:.
6403:.
6395:.
6383:76
6381:.
6377:.
6340:84
6338:.
6334:.
6307:21
6305:.
6301:.
6266:.
6256:69
6254:.
6250:.
6223:.
6215:.
6203:15
6201:.
6169:.
6159:68
6157:.
6153:.
6123:.
6091:.
6081:77
6079:.
6075:.
6055:35
6053:.
6049:.
6023:.
6013:52
5988:.
5980:.
5968:81
5966:.
5960:.
5943:^
5926:.
5905:.
5881:.
5837:.
5795:,
5789:,
5785:,
5764:}}
5760:{{
5743:.
5732:^
5601:.
5535:.
5478:.
5357:1
5264:.
5255:A
5252:.
5188:.
4764:14
4750:28
4490:).
4483:).
4471::
4310:=
4098:.
4058:ln
3954:26
3951:17
3869:37
3856:99
3280:.
3187:10
3174:11
3161:12
3144:10
3048:10
2978:10
2974:28
2921:10
2830:.
2700:,
2684:,
2680:42
2676:,
2672:10
2668:,
2652:,
2648:31
2644:,
2308:13
2295:12
2267:15
2254:13
2241:11
2006:31
2000:30
1994:29
1792:28
1377:).
1298:.
938:64
873:31
867:16
808:28
657:a
326:.
183:,
163:.
80:.
72:,
70:28
68:,
8227:a
8108:e
8101:t
8094:v
7965:)
7961:(
7688:e
7681:t
7674:v
7660:.
7605:.
7551:.
7502:.
7488::
7441:.
7429::
7389:.
7387:.
7327:.
7315::
7292:.
7280::
7274:6
7257:.
7237::
7215:.
7188:.
7186:.
7180::
7121:.
7101::
7091::
7068:.
7037:"
7020:2
7016:M
7006:p
6984:.
6962::
6935:.
6888:.
6860::
6832:.
6810::
6783:.
6761::
6724:.
6696::
6635:.
6604:.
6591::
6561:.
6555::
6526:.
6516:3
6495:.
6489::
6462:.
6426:.
6407::
6399::
6389::
6359:.
6346::
6316:.
6283:.
6270::
6262::
6233:.
6231:.
6219::
6209::
6186:.
6173::
6165::
6135:.
6131::
6125:8
6108:.
6095::
6087::
6031:.
6019::
5996:.
5974::
5937:.
5892:.
5843:.
5819:.
5770:)
5756:.
5726:.
5699:.
5652:.
5605:.
5546:.
5502:.
5488:.
5454:.
5441:p
5426:,
5423:1
5415:p
5411:2
5398:p
5396:2
5390:p
5388:2
5382:p
5380:2
5372:p
5363:p
5361:2
5343:1
5335:p
5331:2
5234:S
5217:n
5213:n
5211:(
5209:Ï
5205:n
5203:(
5201:s
5033:.
5017:)
5012:n
5007:(
5004:o
4994:c
4978:n
4973:c
4963:n
4946:1
4943:+
4938:n
4934:2
4910:)
4907:1
4904:+
4899:n
4895:2
4891:(
4886:1
4880:n
4876:2
4854:.
4845:N
4826:2
4823:=
4820:1
4816:/
4812:1
4809:+
4806:2
4802:/
4798:1
4795:+
4792:4
4788:/
4784:1
4781:+
4778:7
4774:/
4770:1
4767:+
4760:/
4756:1
4753:+
4746:/
4742:1
4731:;
4719:2
4716:=
4711:6
4707:6
4701:2
4695:=
4690:6
4686:6
4683:+
4680:3
4677:+
4674:2
4671:+
4668:1
4662:=
4657:6
4654:6
4649:+
4644:6
4641:3
4636:+
4631:6
4628:2
4623:+
4618:6
4615:1
4610:=
4605:1
4602:1
4597:+
4592:2
4589:1
4584:+
4579:3
4576:1
4571:+
4566:6
4563:1
4548:n
4534:n
4531:2
4528:=
4525:)
4522:n
4519:(
4514:1
4499:N
4477:n
4439:.
4423:3
4420:+
4417:e
4414:8
4411:+
4406:2
4402:e
4398:2
4394:4
4389:2
4382:N
4362:2
4359:+
4356:e
4353:8
4350:+
4345:2
4341:e
4337:2
4331:k
4319:e
4312:e
4307:k
4303:e
4299:1
4296:e
4287:k
4283:e
4279:1
4276:e
4274:(
4271:.
4252:+
4249:t
4246:2
4240:u
4234:4
4230:/
4226:)
4223:1
4217:t
4214:(
4204:u
4200:t
4195:k
4191:e
4187:1
4184:e
4179:.
4177:S
4173:N
4169:S
4164:i
4160:e
4153:N
4140:i
4136:e
4124:i
4120:e
4108:i
4104:e
4095:k
4091:e
4087:1
4084:e
4076:.
4064:2
4048:k
4044:p
4040:1
4035:+
4029:+
4022:2
4018:p
4014:1
4009:+
4002:1
3998:p
3994:1
3989:+
3984:q
3981:1
3969:.
3946:N
3942:2
3934:k
3930:p
3921:3
3917:p
3911:2
3907:p
3901:1
3897:p
3893:q
3883:.
3859:k
3845:k
3841:e
3837:2
3834:+
3828:+
3823:3
3819:e
3815:2
3812:+
3807:2
3803:e
3799:2
3796:+
3791:1
3787:e
3783:2
3780:+
3754:)
3749:1
3746:+
3743:k
3739:2
3730:1
3727:+
3724:k
3720:4
3716:(
3712:2
3705:N
3693:n
3677:.
3672:2
3668:1
3662:k
3649:N
3638:q
3631:k
3627:p
3623:1
3620:p
3616:q
3596:,
3589:k
3585:e
3581:2
3576:k
3572:p
3561:1
3557:e
3553:2
3548:1
3544:p
3534:q
3530:=
3527:N
3511:N
3505:N
3501:N
3497:N
3482:.
3476:6
3471:N
3468:2
3454:.
3439:5
3434:N
3431:2
3406:.
3400:3
3395:N
3392:3
3379:N
3372:N
3368:N
3364:N
3360:N
3354:N
3348:N
3341:N
3303::
3256:2
3248:=
3239:6
3235:2
3231:+
3226:7
3222:2
3218:+
3213:8
3209:2
3205:+
3200:9
3196:2
3192:+
3183:2
3179:+
3170:2
3166:+
3157:2
3149:=
3130:2
3122:=
3115:4
3111:2
3107:+
3102:5
3098:2
3094:+
3089:6
3085:2
3081:+
3076:7
3072:2
3068:+
3063:8
3059:2
3053:=
3034:2
3026:=
3019:2
3015:2
3011:+
3006:3
3002:2
2998:+
2993:4
2989:2
2983:=
2964:2
2956:=
2949:1
2945:2
2941:+
2936:2
2932:2
2926:=
2917:6
2898:p
2893:p
2879:,
2876:)
2873:1
2865:p
2861:2
2857:(
2852:1
2846:p
2842:2
2828:m
2814:)
2811:1
2803:m
2799:2
2795:(
2790:1
2784:m
2780:2
2765:p
2751:)
2748:1
2740:p
2736:2
2732:(
2727:1
2721:p
2717:2
2686:T
2678:T
2670:T
2664:2
2662:T
2654:T
2646:T
2640:7
2638:T
2620:3
2616:/
2612:)
2609:2
2601:p
2597:2
2593:(
2589:T
2582:9
2579:+
2576:1
2573:=
2568:2
2564:)
2561:1
2558:+
2553:p
2549:2
2545:(
2539:)
2536:2
2528:p
2524:2
2520:(
2514:+
2511:1
2508:=
2503:1
2495:p
2491:2
2486:T
2457:3
2449:+
2444:3
2436:+
2431:3
2423:+
2417:+
2412:3
2408:5
2404:+
2399:3
2395:3
2391:+
2386:3
2382:1
2378:=
2363:+
2357:+
2351:+
2345:+
2342:3
2339:+
2336:2
2333:+
2330:1
2327:=
2319:)
2316:1
2304:2
2300:(
2291:2
2287:=
2271:3
2263:+
2258:3
2250:+
2245:3
2237:+
2232:3
2228:9
2224:+
2219:3
2215:7
2211:+
2206:3
2202:5
2198:+
2193:3
2189:3
2185:+
2180:3
2176:1
2172:=
2157:+
2151:+
2145:+
2139:+
2136:3
2133:+
2130:2
2127:+
2124:1
2121:=
2113:)
2110:1
2102:7
2098:2
2094:(
2089:6
2085:2
2081:=
2065:3
2061:7
2057:+
2052:3
2048:5
2044:+
2039:3
2035:3
2031:+
2026:3
2022:1
2018:=
2003:+
1997:+
1991:+
1985:+
1982:3
1979:+
1976:2
1973:+
1970:1
1967:=
1959:)
1956:1
1948:5
1944:2
1940:(
1935:4
1931:2
1927:=
1911:3
1907:3
1903:+
1898:3
1894:1
1890:=
1878:7
1875:+
1872:6
1869:+
1866:5
1863:+
1860:4
1857:+
1854:3
1851:+
1848:2
1845:+
1842:1
1839:=
1831:)
1828:1
1820:3
1816:2
1812:(
1807:2
1803:2
1799:=
1785:,
1782:3
1779:+
1776:2
1773:+
1770:1
1767:=
1759:)
1756:1
1748:2
1744:2
1740:(
1735:1
1731:2
1727:=
1720:6
1694:1
1685:2
1681:1
1678:+
1675:p
1669:2
1645:2
1641:1
1635:p
1629:2
1601:3
1597:1
1594:+
1589:p
1585:2
1554:1
1548:p
1544:2
1523:1
1515:p
1511:2
1486:)
1483:1
1475:p
1471:2
1467:(
1447:)
1444:1
1436:p
1432:2
1428:(
1423:1
1417:p
1413:2
1387:p
1383:p
1365:p
1348:)
1345:1
1337:p
1333:2
1329:(
1324:1
1318:p
1314:2
1278:)
1275:1
1267:p
1263:2
1259:(
1254:1
1248:p
1244:2
1215:1
1207:n
1203:2
1182:)
1179:1
1171:n
1167:2
1163:(
1158:1
1152:n
1148:2
1116:1
1108:p
1104:2
1093:p
1085:p
1071:1
1063:p
1059:2
1048:p
1034:1
1026:p
1022:2
989:1
981:p
977:2
947:=
935:=
932:)
929:1
921:7
917:2
913:(
908:6
904:2
899::
892:7
889:=
886:p
876:=
864:=
861:)
858:1
850:5
846:2
842:(
837:4
833:2
828::
821:5
818:=
815:p
805:=
802:7
796:4
793:=
790:)
787:1
779:3
775:2
771:(
766:2
762:2
757::
750:3
747:=
744:p
737:6
734:=
731:3
725:2
722:=
719:)
716:1
708:2
704:2
700:(
695:1
691:2
686::
679:2
676:=
673:p
655:p
641:,
638:)
635:1
627:p
623:2
619:(
614:1
608:p
604:2
574:1
566:p
562:2
541:)
538:1
530:p
526:2
522:(
517:1
511:p
507:2
483::
439:n
425:1
417:n
413:2
392:)
389:1
381:n
377:2
373:(
368:1
362:n
358:2
302:p
282:1
274:p
270:2
246:q
226:2
222:/
218:)
215:1
212:+
209:q
206:(
203:q
179:(
145:1
120:n
117:2
114:=
111:)
108:n
105:(
100:1
66:6
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
