2473:
1711:
5077:
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:
7763:
3272:
10024:
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
5582:, 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
4749:
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}}}
5279:
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
4094:
4482:
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
4576:
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,
4473:... 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
4289:
669:
3974:
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
4856:
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
4457:
7055:
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.
4744:{\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
8126:
4564:
4392:
3687:
2889:
651:
130:
4940:
2824:
2761:
1457:
1358:
1288:
1192:
551:
402:
5266:
5047:
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,
6868:
5010:
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:
5456:
236:
5373:
4976:
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:
10058:
5303:
3772:
7706:
5466:
1374:
8119:
3309:
1394:
489:
5811:
5765:
5056:
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:
8926:
8112:
5943:
1389:
below 109332539. As of
December 2018, 51 Mersenne primes are known, and therefore 51 even perfect numbers (the largest of which is
3522:
8921:
8936:
8916:
3316:
7752:
7558:
7402:
7229:
5741:
5714:
5667:
5578:
9629:
9209:
7949:
7762:
5133:
6264:
6167:
3336:, and Pace Nielsen has suggested that sufficient study of those numbers may lead to a proof that no odd perfect numbers exist.
8931:
5787:
9715:
4229:
7699:
7593:
4887:. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form
4089:{\displaystyle {\frac {1}{q}}+{\frac {1}{p_{1}}}+{\frac {1}{p_{2}}}+\cdots +{\frac {1}{p_{k}}}>{\frac {\ln k}{2\ln 2}}}
9031:
9381:
8700:
8493:
2660:(after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with
9416:
9386:
9061:
9051:
7903:
6027:
KĂŒhnel, Ullrich (1950). "VerschĂ€rfung der notwendigen
Bedingungen fĂŒr die Existenz von ungeraden vollkommenen Zahlen".
9557:
8971:
8705:
8685:
7588:
5648:
5216:
4488:
9247:
7658:
9411:
10053:
9506:
9129:
8886:
8695:
8677:
8571:
8561:
8551:
7939:
7496:
6769:
6563:
5982:
5129:
9391:
5926:
10048:
9634:
9179:
8800:
8586:
8581:
8576:
8566:
8543:
7924:
7151:
The
Collected Mathematical Papers of James Joseph Sylvester p. 590, tr. from "Sur les nombres dits de Hamilton",
5212:
5098:
4757:
1617:
3700:
1295:
474:
323:
10068:
10063:
8619:
7692:
5816:
5610:
5076:
160:
8876:
7466:
Steuerwald, R. "VerschÀrfung einer notwendigen
Bedingung fĂŒr die Existenz einer ungeraden vollkommenen Zahl".
7397:. Chapman & Hall/CRC Pure and Applied Mathematics. Vol. 201. CRC Press. Problem 7.4.11, p. 428.
4871:
4397:
329:
It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist.
5552:
9745:
9710:
9496:
9406:
9280:
9255:
9164:
9154:
8766:
8748:
8668:
8078:
7944:
7868:
7020:
6866:(1950). "Satze uber Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretisehe Probleme. II".
5323:
5308:
5120:
3329:
37:
1575:
10005:
9275:
9149:
8780:
8556:
8336:
8263:
7929:
7888:
6217:
Zelinsky, Joshua (July 2019). "Upper bounds on the second largest prime factor of an odd perfect number".
5596:
5318:
5107:
4514:
4466:
1663:
5657:
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,
9260:
9114:
9041:
8196:
7858:
7727:
1623:
31:
9969:
9609:
5204:. A positive integer such that every smaller positive integer is a sum of distinct divisors of it is a
4524:
4346:
3654:
2836:
598:
90:
7430:
5895:
5180:
gives various other kinds of numbers. Numbers where the sum is less than the number itself are called
4978:
with a power of two in a similar way to the construction of even perfect numbers from
Mersenne primes.
4890:
2774:
2711:
1407:
1308:
1238:
1142:
501:
352:
9902:
9796:
9760:
9501:
9224:
9204:
9021:
8690:
8478:
8450:
8032:
7934:
7425:
6416:
6279:
6182:
6104:
5395:
5116:
588:
167:
7638:
7606:
7583:
6063:
5806:
5247:
5019:
345:
noted 8128 as early as around AD 100. In modern language, Nicomachus states without proof that
9624:
9488:
9483:
9451:
9214:
9189:
9184:
9159:
9089:
9085:
9016:
8906:
8738:
8534:
8503:
8093:
8088:
7883:
7878:
7863:
7802:
7064:
6931:
5802:
5313:
5293:
5064:
442:
10023:
4988:
3460:
3384:
10027:
9781:
9776:
9690:
9664:
9562:
9541:
9313:
9194:
9144:
9066:
9036:
8976:
8743:
8723:
8654:
8367:
8017:
8012:
7973:
7893:
7873:
7515:
7454:
7340:
7305:
7270:
7262:
7134:
7108:
6901:
6737:
6608:
6432:
6406:
6244:
6226:
6044:
5653:
5378:
5276:
4880:
4168:
4164:
4148:
4132:
3641:
3423:
3325:
2696:) is obtained, always produces the number 1. For example, the digital root of 8128 is 1, because
454:
135:
8911:
6863:
1462:
9921:
9866:
9720:
9695:
9669:
9446:
9124:
9119:
9046:
9026:
9011:
8733:
8715:
8634:
8624:
8609:
8387:
8372:
8053:
7993:
7613:
7554:
7550:
7526:
Riele, H.J.J. "Perfect
Numbers and Aliquot Sequences" in H.W. Lenstra and R. Tijdeman (eds.):
7398:
7392:
7225:
7126:
7073:
6989:
6940:
6885:
6837:
6788:
6721:
6001:
5851:
5781:
5737:
5731:
5710:
5704:
5663:
5583:
5425:
5050:
4884:
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:
7375:
7219:
5687:
5345:
4948:
1538:
1505:
1197:
1098:
1053:
1016:
971:
556:
407:
264:
9957:
9750:
9336:
9308:
9298:
9290:
9174:
9139:
9134:
9101:
8795:
8758:
8649:
8644:
8639:
8629:
8601:
8488:
8440:
8435:
8392:
8331:
8083:
8058:
7978:
7964:
7898:
7782:
7742:
7564:
7542:
7505:
7446:
7332:
7297:
7254:
7197:
7118:
6979:
6877:
6827:
6778:
6713:
6572:
6506:
6424:
6363:
6287:
6236:
6190:
6148:
6112:
6036:
6009:
5991:
5879:
5536:
5269:
5241:
5205:
5181:
5169:
5160:
3333:
3284:
1569:
54:
7085:
7001:
6952:
6897:
6849:
6800:
6733:
9933:
9822:
9755:
9681:
9604:
9578:
9396:
9109:
8966:
8901:
8871:
8861:
8856:
8522:
8430:
8377:
8221:
8161:
8068:
8063:
7988:
7982:
7919:
7817:
7807:
7737:
7568:
7081:
6997:
6948:
6893:
6845:
6796:
6729:
6013:
5930:
5201:
5197:
5185:
5089:
6704:
McDaniel, Wayne L. (1970). "The non-existence of odd perfect numbers of a certain form".
4506:
28 is also the only even perfect number that is a sum of two positive cubes of integers (
7099:
Yamada, Tomohiro (2019). "A new upper bound for odd perfect numbers of a special form".
6420:
6283:
6186:
6108:
6089:
4883:; 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
9938:
9806:
9791:
9655:
9619:
9594:
9470:
9441:
9426:
9199:
8896:
8851:
8728:
8326:
8321:
8316:
8288:
8273:
8186:
8171:
8149:
8136:
8073:
8027:
7837:
7827:
7797:
5519:
5298:
5244:
associated with a perfect number is a constant sequence. All perfect numbers are also
5177:
4863:
The number of divisors of a perfect number (whether even or odd) must be even, because
4484:
3321:
1232:
1228:
1006:
1002:
462:
319:
315:
297:
241:
7684:
6984:
6967:
5390:, and both 23 and 89 yield a remainder of 1 when divided by 22. Furthermore, whenever
2912:
10042:
9861:
9845:
9786:
9740:
9436:
9421:
9331:
9056:
8614:
8483:
8445:
8402:
8283:
8268:
8258:
8216:
8206:
8181:
8022:
7822:
7812:
7792:
7543:
7458:
7371:
7344:
7309:
7274:
7138:
6905:
6741:
6048:
5875:
5683:
5532:
5080:
3876:{\displaystyle \alpha +2e_{1}+2e_{2}+2e_{3}+\cdots +2e_{k}\geq {\frac {99k-224}{37}}}
1010:
77:
46:
7650:
6783:
6761:
6577:
6558:
6368:
6248:
6139:
Konyagin, Sergei; Acquaah, Peter (2012). "On Prime
Factors of Odd Perfect Numbers".
5996:
9897:
9886:
9801:
9639:
9614:
9531:
9431:
9401:
9376:
9360:
9265:
9232:
8981:
8955:
8866:
8805:
8382:
8278:
8211:
8191:
8166:
8037:
7954:
7832:
7777:
7747:
7616:
7016:
6436:
5854:
5281:
5142:
5057:
4943:
2693:
1398:
1305:
distributed computing project has shown that the first 48 even perfect numbers are
658:
73:
7486:
Nankar, M.L.: "History of perfect numbers," Ganita
Bharati 1, no. 1â2 (1979), 7â8.
6428:
6315:
6292:
6195:
6117:
5520:"A proof that all even perfect numbers are a power of two times a Mersenne prime"
5196:. A pair of numbers which are the sum of each other's proper divisors are called
9856:
9731:
9536:
9000:
8891:
8846:
8841:
8591:
8498:
8397:
8226:
8201:
8176:
7666:
6923:
6594:"On inequalities involving counts of the prime factors of an odd perfect number"
6593:
6461:
6168:"The second largest prime divisor of an odd perfect number exceeds ten thousand"
259:
84:
69:
6348:
1381:
Three higher perfect numbers have also been discovered, namely those for which
9993:
9974:
9270:
8881:
7670:
7017:"Some results concerning the non-existence of odd perfect numbers of the form
6881:
6832:
6815:
6511:
6494:
6391:
6265:"The third largest prime divisor of an odd perfect number exceeds one hundred"
6240:
6152:
5760:
5193:
1132:
458:
342:
65:
8104:
7450:
7323:
Kanold, HJ (1956). "Eine
Bemerkung šuber die Menge der vollkommenen zahlen".
7130:
7077:
6993:
6944:
6889:
6841:
6792:
6725:
6005:
1091:
is not a prime number. In fact, Mersenne primes are very rare: of the primes
9599:
9526:
9518:
9323:
9237:
8355:
7787:
7641:
A projected distributed computing project to search for odd perfect numbers.
7621:
6527:
5859:
1291:
322:
proved that all even perfect numbers are of this form. This is known as the
7245:
Jones, Chris; Lord, Nick (1999). "Characterising non-trapezoidal numbers".
7122:
6636:
5595:
Commentary on the Gospel of John 28.1.1â4, with further references in the
4195:
must be smaller than an effectively computable constant depending only on
17:
9700:
7437:
Kanold, H.-J. (1941). "Untersuchungen ĂŒber ungerade vollkommene Zahlen".
5974:
5612:
The Reception of Philonic Arithmological Exegesis in Didymus the Blind's
595:
For example, the first four perfect numbers are generated by the formula
4521:
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.
9705:
9364:
7732:
7519:
7336:
7301:
7266:
6717:
6040:
5922:
3298:
478:
58:
7202:
7185:
6755:
6753:
6751:
6411:
5637:
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:
7510:
7491:
7288:
Hornfeck, B (1955). "Zur Dichte der Menge der vollkommenen zahlen".
7258:
4218:) = (1, ..., 1, 2, ..., 2) with
7433:
for a translation and discussion of this proposition and its proof.
7113:
6613:
6231:
3962:{\displaystyle qp_{1}p_{2}p_{3}\cdots p_{k}<2N^{\frac {17}{26}}}
8003:
6314:
Bibby, Sean; Vyncke, Pieter; Zelinsky, Joshua (23 November 2021).
5075:
4870:
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:
7630:
5623:
Society of Biblical Literature National Meeting, Atlanta, Georgia
4102:
Furthermore, several minor results are known about the exponents
3601:{\displaystyle N=q^{\alpha }p_{1}^{2e_{1}}\cdots p_{k}^{2e_{k}},}
5736:. Washington: Mathematical Association of America. p. 132.
5706:
Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning
5639:(Dordrecht: Kluwer Academic Publishers, 1994), pp. 328â329.
9991:
9955:
9919:
9883:
9843:
9468:
9357:
9083:
8998:
8953:
8830:
8520:
8467:
8419:
8353:
8305:
8243:
8147:
8108:
7688:
6462:"On the Total Number of Prime Factors of an Odd Perfect Number"
6392:"Odd perfect numbers have at least nine distinct prime factors"
7601:
7394:
Number Theory: An Introduction to Pure and Applied Mathematics
6968:"A new result concerning the structure of odd perfect numbers"
6349:"Odd perfect numbers, Diophantine equations, and upper bounds"
5944:"Mathematicians Open a New Front on an Ancient Number Problem"
7380:. Washington: Carnegie Institution of Washington. p. 25.
6316:"On the Third Largest Prime Divisor of an Odd Perfect Number"
5692:. Washington: Carnegie Institution of Washington. p. 10.
174:
7644:
6917:
6915:
5884:. Washington: Carnegie Institution of Washington. p. 6.
5541:. Washington: Carnegie Institution of Washington. p. 4.
5253:
1572:. Furthermore, each even perfect number except for 6 is the
7633:
6924:"Extensions of some results concerning odd perfect numbers"
6760:
Fletcher, S. Adam; Nielsen, Pace P.; Ochem, Pascal (2012).
5829:
5495:
5470:
2891:
every even perfect number is represented in binary form as
1369:
1290:
will yield all the even perfect numbers. Thus, there is a
6559:"On the number of prime factors of an odd perfect number"
5733:
Mathematical Treks: From Surreal Numbers to Magic Circles
4310:) ≠ (1, ..., 1, 3), (1, ..., 1, 5), (1, ..., 1, 6).
41:
Illustration of the perfect number status of the number 6
7492:"A Lower Bound for the set of odd Perfect Prime Numbers"
6820:
Journal of the Australian Mathematical Society, Series A
6685:
Suryanarayana, D. (1963). "On Odd Perfect Numbers II".
5968:
5966:
5964:
1227:
is prime (though he stated this somewhat differently),
6090:"Odd perfect numbers have a prime factor exceeding 10"
4284:{\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
7224:, John Wiley & Sons, Section 2.3, Exercise 2(6),
7023:
5428:
5348:
5250:
5022:
4991:
4951:
4893:
4760:
4579:
4527:
4400:
4349:
4232:
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
7535:
Prime Numbers and Computer Methods for Factorisation
6495:"Improved upper bounds for odd multiperfect numbers"
9815:
9769:
9729:
9680:
9654:
9587:
9571:
9550:
9517:
9482:
9322:
9289:
9246:
9223:
9100:
8788:
8779:
8757:
8714:
8676:
8667:
8600:
8542:
8533:
8046:
8002:
7963:
7912:
7846:
7770:
7720:
7221:
Computational Number Theory and Modern Cryptography
6816:"On the largest component of an odd perfect number"
5755:
5753:
7165:Makowski, A. (1962). "Remark on perfect numbers".
7049:
5450:
5367:
5260:
5041:
5004:
4970:
4934:
4850:
4743:
4558:
4451:
4386:
4283:
4088:
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:
6972:Proceedings of the American Mathematical Society
6687:Proceedings of the American Mathematical Society
6666:Cohen, Graeme (1978). "On odd perfect numbers".
6455:
6453:
5709:. 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
7439:Journal fĂŒr die Reine und Angewandte Mathematik
7186:"On a remark of Makowski about perfect numbers"
6869:Journal fĂŒr die reine und angewandte Mathematik
6499:Bulletin of the Australian Mathematical Society
5060:of every even perfect number other than 6 is 1.
4471:
27:Integer equal to the sum of its proper divisors
7530:, Vol. 154, Amsterdam, 1982, pp. 141â157.
7253:(497). The Mathematical Association: 262â263.
8120:
7700:
7015:McDaniel, Wayne L.; Hagis, Peter Jr. (1975).
6966:Hagis, Peter Jr.; McDaniel, Wayne L. (1972).
4187:+1 have a prime factor in a given finite set
195:also proved a formation rule (IX.36) whereby
8:
7634:sequence A000396 (Perfect numbers)
5304:List of Mersenne primes and perfect numbers
5184:, and where it is greater than the number,
4851:{\displaystyle 1/28+1/14+1/7+1/4+1/2+1/1=2}
9988:
9952:
9916:
9880:
9840:
9514:
9479:
9465:
9354:
9097:
9080:
8995:
8950:
8827:
8785:
8673:
8539:
8530:
8517:
8464:
8421:Possessing a specific set of other numbers
8416:
8350:
8302:
8240:
8144:
8127:
8113:
8105:
7707:
7693:
7685:
7541:SĂĄndor, Jozsef; Crstici, Borislav (2004).
5463:= 11, 23, 83, 131, 179, 191, 239, 251, ...
5200:, 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?
7509:
7201:
7112:
7038:
7028:
7022:
6983:
6831:
6782:
6612:
6576:
6510:
6410:
6367:
6291:
6230:
6194:
6116:
5995:
5975:"Odd perfect numbers are greater than 10"
5433:
5427:
5353:
5347:
5252:
5251:
5249:
5029:
5021:
4995:
4990:
4956:
4950:
4917:
4898:
4892:
4834:
4820:
4806:
4792:
4778:
4764:
4759:
4717:
4684:
4671:
4658:
4645:
4632:
4619:
4606:
4593:
4580:
4578:
4532:
4526:
4495:The only even perfect number of the form
4424:
4416:
4411:
4399:
4363:
4348:
4274:
4248:
4231:
4057:
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:
7377:History of the Theory of Numbers, Vol. I
6528:"An upper bound for odd perfect numbers"
5896:"The oldest open problem in mathematics"
5881:History of the Theory of Numbers, Vol. I
5689:History of the Theory of Numbers, Vol. I
5538:History of the Theory of Numbers, Vol. I
4981:The number of perfect numbers less than
4507:
4500:
4452:{\displaystyle N<2^{4^{2e^{2}+8e+3}}}
2826:for odd integer (not necessarily prime)
7549:. Dordrecht: Kluwer Academic. pp.
7153:Compte Rendu de l'Association Française
6762:"Sieve methods for odd perfect numbers"
6635:Pomerance, Carl; Luca, Florian (2010).
5812:MacTutor History of Mathematics Archive
5487:
5335:
4341: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)
7602:Perfect, amicable and sociable numbers
7528:Computational Methods in Number Theory
7050:{\displaystyle p^{\alpha }M^{2\beta }}
6922:Cohen, G. L.; Williams, R. J. (1985).
6219:International Journal of Number Theory
6141:International Journal of Number Theory
6064:"On the Form of an Odd Perfect Number"
5779:
4171:5), 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
5211: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:
7645:Great Internet Mersenne Prime Search
7607:Perfect numbers â History and Theory
6637:"On the radical of a perfect number"
6592:Graeme Clayton, Cody Hansen (2023).
6557:Ochem, Pascal; Rao, Michaël (2014).
6493:Chen, Yong-Gao; Tang, Cui-E (2014).
5973:Ochem, Pascal; Rao, Michaël (2012).
5766:Great Internet Mersenne Prime Search
5016:> 0 is a constant. In fact it is
4517:of the divisors of a perfect number
4191:, 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}
7715:Divisibility-based sets of integers
5414: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
10059:Unsolved problems in number theory
6460:Zelinsky, Joshua (3 August 2021).
5942: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:
7753:Fundamental theorem of arithmetic
6985:10.1090/S0002-9939-1972-0292740-5
5807:"Abu Ali al-Hasan ibn al-Haytham"
5659:History of Mathematics: Volume II
4879:The even perfect numbers are not
4559:{\displaystyle \sigma _{1}(n)=2n}
4387:{\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}
10022:
9630:Perfect digit-to-digit invariant
7761:
4935:{\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)}
7429:, Book IX, Proposition 36. See
6784:10.1090/S0025-5718-2011-02576-7
6641:New York Journal of Mathematics
6578:10.1090/S0025-5718-2013-02776-7
6369:10.1090/S0025-5718-2015-02941-X
6071:Australian Mathematical Gazette
5997:10.1090/S0025-5718-2012-02563-4
5662:. 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
7361:Texeira J. VIII (1886), 11â16.
7155:(Toulouse, 1887), pp. 164â168.
5261:{\displaystyle {\mathcal {S}}}
5042:{\displaystyle o({\sqrt {n}})}
5036:
5026:
4929:
4910:
4544:
4538:
4245:
4233:
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:
8469:Expressible via specific sums
7359:Note sur les nombres parfaits
6429:10.1090/S0025-5718-07-01990-4
6293:10.1090/S0025-5718-99-01127-8
6196:10.1090/S0025-5718-99-01126-6
6118:10.1090/S0025-5718-08-02050-9
5188:. These terms, together with
3647:The smallest prime factor of
7545:Handbook of number theory II
5005:{\displaystyle c{\sqrt {n}}}
4867: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
9558:Multiplicative digital root
7589:Encyclopedia of Mathematics
5649:Bayerische Staatsbibliothek
5557:www-groups.dcs.st-and.ac.uk
5217:restricted divisor function
4942:formed as the product of a
4566:, and divide both sides by
4489:strong law of small numbers
4175:must lie between 10 and 10.
3447:{\displaystyle {\sqrt{2N}}}
1404:As well as having the form
152:{\displaystyle \sigma _{1}}
10085:
7497:Mathematics of Computation
7184:Gallardo, Luis H. (2010).
6770:Mathematics of Computation
6564:Mathematics of Computation
6399:Mathematics of Computation
6356:Mathematics of Computation
6272:Mathematics of Computation
6175:Mathematics of Computation
6097:Mathematics of Computation
5983:Mathematics of Computation
5786:: CS1 maint: url-status (
5609:Rogers, Justin M. (2015).
5586:based on a Mersenne prime.
5579:Introduction to Arithmetic
968:Prime numbers of the form
472:
175:
29:
10018:
10001:
9987:
9965:
9951:
9929:
9915:
9893:
9879:
9852:
9839:
9635:Perfect digital invariant
9478:
9464:
9372:
9353:
9210:Superior highly composite
9096:
9079:
9007:
8994:
8962:
8949:
8837:
8826:
8529:
8516:
8474:
8463:
8426:
8415:
8363:
8349:
8312:
8301:
8254:
8239:
8157:
8143:
7950:Superior highly composite
7759:
6882:10.1515/crll.1950.188.129
6833:10.1017/S1446788700028251
6526:Nielsen, Pace P. (2003).
6512:10.1017/S0004972713000488
6390:Nielsen, Pace P. (2007).
6347:Nielsen, Pace P. (2015).
6241:10.1142/S1793042119500659
6153:10.1142/S1793042112500935
6088:Goto, T; Ohno, Y (2008).
6029:Mathematische Zeitschrift
5761:"GIMPS Milestones Report"
5599:edition: vol. 385, 58â61.
5134:superior highly composite
1618:centered nonagonal number
1491:{\displaystyle (2^{p}-1)}
1292:one-to-one correspondence
1013:and perfect numbers. For
9248:Euler's totient function
9032:EulerâJacobi pseudoprime
8307:Other polynomial numbers
7847:Constrained divisor sums
7451:10.1515/crll.1941.183.98
7247:The Mathematical Gazette
5817:University of St Andrews
5451:{\displaystyle 2^{p}-1,}
5192:itself, come from Greek
4178: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
9062:SomerâLucas pseudoprime
9052:LucasâCarmichael number
8887:Lazy caterer's sequence
7659:"8128: Perfect Numbers"
7653:, math forum at Drexel.
7468:S.-B. Bayer. Akad. Wiss
7101:Colloquium Mathematicum
5368:{\displaystyle 2^{p}-1}
5324:Harmonic divisor number
5309:Multiply perfect number
4971:{\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
8937:WedderburnâEtherington
8337:Lucky numbers of Euler
7051:
5458:which is the case for
5452:
5388:2 â 1 = 2047 = 23 Ă 89
5369:
5319:Unitary perfect number
5262:
5173:
5083:of numbers under 100:
5043:
5006:
4972:
4936:
4852:
4745:
4560:
4499: + 1 is 28 (
4475:
4453:
4388:
4285:
4090:
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:
9225:Prime omega functions
9042:Frobenius pseudoprime
8832:Combinatorial numbers
8701:Centered dodecahedral
8494:Primary pseudoperfect
7728:Integer factorization
7391:Redmond, Don (1996).
7218:Yan, Song Y. (2012),
7123:10.4064/cm7339-3-2018
7052:
6814:Cohen, G. L. (1987).
6706:Archiv der Mathematik
6263:Iannucci, DE (2000).
6166:Iannucci, DE (1999).
5614:Commentary on Genesis
5453:
5370:
5268:-perfect numbers, or
5263:
5079:
5044:
5007:
4973:
4937:
4872:Ore's harmonic number
4853:
4746:
4561:
4454:
4389:
4286:
4091:
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)
9684:-composition related
9484:Arithmetic functions
9086:Arithmetic functions
9022:Elliptic pseudoprime
8706:Centered icosahedral
8686:Centered tetrahedral
7431:D.E. Joyce's website
7021:
6864:Kanold, Hans-Joachim
5803:Robertson, Edmund F.
5730:Peterson, I (2002).
5703:Pickover, C (2001).
5426:
5422:will be a factor of
5396:Sophie Germain prime
5346:
5314:Superperfect numbers
5248:
5067:perfect number is 6.
5020:
4989:
4949:
4891:
4758:
4577:
4525:
4398:
4347:
4230:
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:
9610:Kaprekar's constant
9130:Colossally abundant
9017:Catalan pseudoprime
8917:SchröderâHipparchus
8696:Centered octahedral
8572:Centered heptagonal
8562:Centered pentagonal
8552:Centered triangular
8152:and related numbers
7940:Colossally abundant
7771:Factorization forms
7537:, Birkhauser, 1985.
7065:Fibonacci Quarterly
6932:Fibonacci Quarterly
6668:Fibonacci Quarterly
6421:2007MaCom..76.2109N
6284:2000MaCom..69..867I
6187:1999MaCom..68.1749I
6109:2008MaCom..77.1859G
6062:Roberts, T (2008).
5801:O'Connor, John J.;
5597:Sources Chrétiennes
5294:Hyperperfect number
5130:Colossally abundant
4881: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
10028:Mathematics portal
9970:Aronson's sequence
9716:SmarandacheâWellin
9473:-dependent numbers
9180:Primitive abundant
9067:Strong pseudoprime
9057:Perrin pseudoprime
9037:Fermat pseudoprime
8977:Wolstenholme prime
8801:Squared triangular
8587:Centered decagonal
8582:Centered nonagonal
8577:Centered octagonal
8567:Centered hexagonal
7925:Primitive abundant
7913:With many divisors
7614:Weisstein, Eric W.
7490:Hagis, P. (1973).
7337:10.1007/BF01350108
7302:10.1007/BF01901120
7047:
6777:(279): 1753?1776.
6718:10.1007/BF01220877
6571:(289): 2435â2439.
6405:(260): 2109â2126.
6362:(295): 2549â2567.
6181:(228): 1749â1760.
6103:(263): 1859â1868.
6041:10.1007/BF02230691
5990:(279): 1869â1877.
5929:2006-12-29 at the
5852:Weisstein, Eric W.
5654:David Eugene Smith
5448:
5406:is also primeâand
5365:
5277:semiperfect number
5258:
5174:
5099:Primitive abundant
5039:
5002:
4968:
4932:
4885:triangular numbers
4848:
4741:
4556:
4449:
4384:
4281:
4086:
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:
10054:Integer sequences
10036:
10035:
10014:
10013:
9983:
9982:
9947:
9946:
9911:
9910:
9875:
9874:
9835:
9834:
9831:
9830:
9650:
9649:
9460:
9459:
9349:
9348:
9345:
9344:
9291:Aliquot sequences
9102:Divisor functions
9075:
9074:
9047:Lucas pseudoprime
9027:Euler pseudoprime
9012:Carmichael number
8990:
8989:
8945:
8944:
8822:
8821:
8818:
8817:
8814:
8813:
8775:
8774:
8663:
8662:
8620:Square triangular
8512:
8511:
8459:
8458:
8411:
8410:
8345:
8344:
8297:
8296:
8235:
8234:
8102:
8101:
5651:, Clm 14908. See
5584:triangular number
5553:"Perfect numbers"
5518:Caldwell, Chris,
5375:are congruent to
5270:Granville numbers
5051:little-o notation
5034:
5000:
4733:
4712:
4679:
4666:
4653:
4640:
4627:
4614:
4601:
4588:
4301:, ...,
4279:
4209:, ...,
4109:, ...,
4084:
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
10076:
10049:Divisor function
10026:
9989:
9958:Natural language
9953:
9917:
9885:Generated via a
9881:
9841:
9746:Digit-reassembly
9711:Self-descriptive
9515:
9480:
9466:
9417:LucasâCarmichael
9407:Harmonic divisor
9355:
9281:Sparsely totient
9256:Highly cototient
9165:Multiply perfect
9155:Highly composite
9098:
9081:
8996:
8951:
8932:Telephone number
8828:
8786:
8767:Square pyramidal
8749:Stella octangula
8674:
8540:
8531:
8523:Figurate numbers
8518:
8465:
8417:
8351:
8303:
8241:
8145:
8129:
8122:
8115:
8106:
8079:Harmonic divisor
7965:Aliquot sequence
7945:Highly composite
7869:Multiply perfect
7765:
7743:Divisor function
7709:
7702:
7695:
7686:
7681:
7679:
7678:
7669:. Archived from
7632:
7627:
7626:
7617:"Perfect Number"
7597:
7584:"Perfect number"
7572:
7548:
7523:
7513:
7504:(124): 951â953.
7475:
7462:
7410:
7408:
7388:
7382:
7381:
7368:
7362:
7355:
7349:
7348:
7320:
7314:
7313:
7285:
7279:
7278:
7242:
7236:
7234:
7215:
7209:
7207:
7205:
7181:
7175:
7174:
7162:
7156:
7149:
7143:
7142:
7116:
7096:
7090:
7089:
7061:
7056:
7054:
7053:
7048:
7046:
7045:
7033:
7032:
7012:
7006:
7005:
6987:
6963:
6957:
6956:
6928:
6919:
6910:
6909:
6860:
6854:
6853:
6835:
6811:
6805:
6804:
6786:
6766:
6757:
6746:
6745:
6701:
6695:
6694:
6682:
6676:
6675:
6663:
6657:
6656:
6654:
6652:
6632:
6626:
6625:
6623:
6621:
6616:
6598:
6589:
6583:
6582:
6580:
6554:
6548:
6547:
6545:
6543:
6523:
6517:
6516:
6514:
6490:
6484:
6483:
6481:
6479:
6466:
6457:
6448:
6447:
6445:
6443:
6414:
6396:
6387:
6381:
6380:
6378:
6376:
6371:
6353:
6344:
6338:
6337:
6335:
6333:
6320:
6311:
6305:
6304:
6302:
6300:
6295:
6278:(230): 867â879.
6269:
6260:
6254:
6252:
6234:
6225:(6): 1183â1189.
6214:
6208:
6207:
6205:
6203:
6198:
6172:
6163:
6157:
6156:
6147:(6): 1537â1540.
6136:
6130:
6129:
6127:
6125:
6120:
6094:
6085:
6079:
6078:
6068:
6059:
6053:
6052:
6024:
6018:
6017:
5999:
5979:
5970:
5959:
5958:
5956:
5954:
5939:
5933:
5920:
5914:
5913:
5911:
5909:
5900:
5892:
5886:
5885:
5872:
5866:
5865:
5864:
5855:"Perfect Number"
5847:
5841:
5840:
5838:
5837:
5826:
5820:
5819:
5798:
5792:
5791:
5785:
5777:
5775:
5773:
5757:
5748:
5747:
5727:
5721:
5720:
5700:
5694:
5693:
5680:
5674:
5673:
5646:
5640:
5633:
5627:
5626:
5620:
5606:
5600:
5593:
5587:
5574:
5568:
5567:
5565:
5563:
5549:
5543:
5542:
5529:
5523:
5516:
5510:
5509:
5507:
5506:
5496:"A000396 - OEIS"
5492:
5475:
5473:
5464:
5462:
5457:
5455:
5454:
5449:
5438:
5437:
5421:
5413:
5405:
5393:
5389:
5385:
5374:
5372:
5371:
5366:
5358:
5357:
5340:
5267:
5265:
5264:
5259:
5257:
5256:
5242:aliquot sequence
5239:
5206:practical number
5167:
5158:
5149:
5140:
5127:
5121:highly composite
5114:
5105:
5096:
5087:
5072:Related concepts
5048:
5046:
5045:
5040:
5035:
5030:
5011:
5009:
5008:
5003:
5001:
4996:
4977:
4975:
4974:
4969:
4961:
4960:
4941:
4939:
4938:
4933:
4922:
4921:
4909:
4908:
4857:
4855:
4854:
4849:
4838:
4824:
4810:
4796:
4782:
4768:
4754:For 28, we have
4750:
4748:
4747:
4742:
4734:
4729:
4718:
4713:
4708:
4685:
4680:
4672:
4667:
4659:
4654:
4646:
4641:
4633:
4628:
4620:
4615:
4607:
4602:
4594:
4589:
4581:
4565:
4563:
4562:
4557:
4537:
4536:
4458:
4456:
4455:
4450:
4448:
4447:
4446:
4445:
4429:
4428:
4393:
4391:
4390:
4385:
4368:
4367:
4334:
4290:
4288:
4287:
4282:
4280:
4275:
4252:
4163: ⥠1 (
4147: ⥠2 (
4131: ⥠1 (
4095:
4093:
4092:
4087:
4085:
4083:
4069:
4058:
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:
10084:
10083:
10079:
10078:
10077:
10075:
10074:
10073:
10069:Perfect numbers
10064:Mersenne primes
10039:
10038:
10037:
10032:
10010:
10006:Strobogrammatic
9997:
9979:
9961:
9943:
9925:
9907:
9889:
9871:
9848:
9827:
9811:
9770:Divisor-related
9765:
9725:
9676:
9646:
9583:
9567:
9546:
9513:
9486:
9474:
9456:
9368:
9367:related numbers
9341:
9318:
9285:
9276:Perfect totient
9242:
9219:
9150:Highly abundant
9092:
9071:
9003:
8986:
8958:
8941:
8927:Stirling second
8833:
8810:
8771:
8753:
8710:
8659:
8596:
8557:Centered square
8525:
8508:
8470:
8455:
8422:
8407:
8359:
8358:defined numbers
8341:
8308:
8293:
8264:Double Mersenne
8250:
8231:
8153:
8139:
8137:natural numbers
8133:
8103:
8098:
8042:
7998:
7959:
7930:Highly abundant
7908:
7889:Unitary perfect
7842:
7766:
7757:
7738:Unitary divisor
7716:
7713:
7676:
7674:
7657:Grimes, James.
7656:
7651:Perfect Numbers
7612:
7611:
7582:
7579:
7561:
7540:
7511:10.2307/2005530
7489:
7483:
7481:Further reading
7478:
7465:
7445:(183): 98â109.
7436:
7419:
7414:
7413:
7405:
7390:
7389:
7385:
7370:
7369:
7365:
7356:
7352:
7322:
7321:
7317:
7287:
7286:
7282:
7259:10.2307/3619053
7244:
7243:
7239:
7232:
7217:
7216:
7212:
7183:
7182:
7178:
7164:
7163:
7159:
7150:
7146:
7098:
7097:
7093:
7059:
7034:
7024:
7019:
7018:
7014:
7013:
7009:
6965:
6964:
6960:
6926:
6921:
6920:
6913:
6862:
6861:
6857:
6813:
6812:
6808:
6764:
6759:
6758:
6749:
6703:
6702:
6698:
6684:
6683:
6679:
6665:
6664:
6660:
6650:
6648:
6634:
6633:
6629:
6619:
6617:
6596:
6591:
6590:
6586:
6556:
6555:
6551:
6541:
6539:
6525:
6524:
6520:
6492:
6491:
6487:
6477:
6475:
6464:
6459:
6458:
6451:
6441:
6439:
6394:
6389:
6388:
6384:
6374:
6372:
6351:
6346:
6345:
6341:
6331:
6329:
6318:
6313:
6312:
6308:
6298:
6296:
6267:
6262:
6261:
6257:
6216:
6215:
6211:
6201:
6199:
6170:
6165:
6164:
6160:
6138:
6137:
6133:
6123:
6121:
6092:
6087:
6086:
6082:
6066:
6061:
6060:
6056:
6026:
6025:
6021:
5977:
5972:
5971:
5962:
5952:
5950:
5948:Quanta Magazine
5941:
5940:
5936:
5931:Wayback Machine
5921:
5917:
5907:
5905:
5898:
5894:
5893:
5889:
5874:
5873:
5869:
5850:
5849:
5848:
5844:
5835:
5833:
5828:
5827:
5823:
5800:
5799:
5795:
5778:
5771:
5769:
5759:
5758:
5751:
5744:
5729:
5728:
5724:
5717:
5702:
5701:
5697:
5682:
5681:
5677:
5670:
5652:
5647:
5643:
5635:Roshdi Rashed,
5634:
5630:
5618:
5608:
5607:
5603:
5594:
5590:
5575:
5571:
5561:
5559:
5551:
5550:
5546:
5531:
5530:
5526:
5517:
5513:
5504:
5502:
5494:
5493:
5489:
5484:
5479:
5478:
5465:
5460:
5459:
5429:
5424:
5423:
5415:
5407:
5399:
5391:
5387:
5386:. For example,
5376:
5349:
5344:
5343:
5342:All factors of
5341:
5337:
5332:
5290:
5246:
5245:
5219:
5178:proper divisors
5172:
5165:
5163:
5156:
5154:
5147:
5145:
5138:
5136:
5125:
5123:
5112:
5110:
5108:Highly abundant
5103:
5101:
5094:
5092:
5085:
5074:
5018:
5017:
4987:
4986:
4952:
4947:
4946:
4913:
4894:
4889:
4888:
4756:
4755:
4719:
4686:
4575:
4574:
4573:For 6, we have
4528:
4523:
4522:
4480:
4420:
4412:
4407:
4396:
4395:
4359:
4345:
4344:
4329:
4320:
4314:
4309:
4300:
4228:
4227:
4217:
4208:
4186:
4162:
4146:
4130:
4117:
4108:
4070:
4059:
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:
15:
12:
11:
5:
10082:
10080:
10072:
10071:
10066:
10061:
10056:
10051:
10041:
10040:
10034:
10033:
10031:
10030:
10019:
10016:
10015:
10012:
10011:
10009:
10008:
10002:
9999:
9998:
9992:
9985:
9984:
9981:
9980:
9978:
9977:
9972:
9966:
9963:
9962:
9956:
9949:
9948:
9945:
9944:
9942:
9941:
9939:Sorting number
9936:
9934:Pancake number
9930:
9927:
9926:
9920:
9913:
9912:
9909:
9908:
9906:
9905:
9900:
9894:
9891:
9890:
9884:
9877:
9876:
9873:
9872:
9870:
9869:
9864:
9859:
9853:
9850:
9849:
9846:Binary numbers
9844:
9837:
9836:
9833:
9832:
9829:
9828:
9826:
9825:
9819:
9817:
9813:
9812:
9810:
9809:
9804:
9799:
9794:
9789:
9784:
9779:
9773:
9771:
9767:
9766:
9764:
9763:
9758:
9753:
9748:
9743:
9737:
9735:
9727:
9726:
9724:
9723:
9718:
9713:
9708:
9703:
9698:
9693:
9687:
9685:
9678:
9677:
9675:
9674:
9673:
9672:
9661:
9659:
9656:P-adic numbers
9652:
9651:
9648:
9647:
9645:
9644:
9643:
9642:
9632:
9627:
9622:
9617:
9612:
9607:
9602:
9597:
9591:
9589:
9585:
9584:
9582:
9581:
9575:
9573:
9572:Coding-related
9569:
9568:
9566:
9565:
9560:
9554:
9552:
9548:
9547:
9545:
9544:
9539:
9534:
9529:
9523:
9521:
9512:
9511:
9510:
9509:
9507:Multiplicative
9504:
9493:
9491:
9476:
9475:
9471:Numeral system
9469:
9462:
9461:
9458:
9457:
9455:
9454:
9449:
9444:
9439:
9434:
9429:
9424:
9419:
9414:
9409:
9404:
9399:
9394:
9389:
9384:
9379:
9373:
9370:
9369:
9358:
9351:
9350:
9347:
9346:
9343:
9342:
9340:
9339:
9334:
9328:
9326:
9320:
9319:
9317:
9316:
9311:
9306:
9301:
9295:
9293:
9287:
9286:
9284:
9283:
9278:
9273:
9268:
9263:
9261:Highly totient
9258:
9252:
9250:
9244:
9243:
9241:
9240:
9235:
9229:
9227:
9221:
9220:
9218:
9217:
9212:
9207:
9202:
9197:
9192:
9187:
9182:
9177:
9172:
9167:
9162:
9157:
9152:
9147:
9142:
9137:
9132:
9127:
9122:
9117:
9115:Almost perfect
9112:
9106:
9104:
9094:
9093:
9084:
9077:
9076:
9073:
9072:
9070:
9069:
9064:
9059:
9054:
9049:
9044:
9039:
9034:
9029:
9024:
9019:
9014:
9008:
9005:
9004:
8999:
8992:
8991:
8988:
8987:
8985:
8984:
8979:
8974:
8969:
8963:
8960:
8959:
8954:
8947:
8946:
8943:
8942:
8940:
8939:
8934:
8929:
8924:
8922:Stirling first
8919:
8914:
8909:
8904:
8899:
8894:
8889:
8884:
8879:
8874:
8869:
8864:
8859:
8854:
8849:
8844:
8838:
8835:
8834:
8831:
8824:
8823:
8820:
8819:
8816:
8815:
8812:
8811:
8809:
8808:
8803:
8798:
8792:
8790:
8783:
8777:
8776:
8773:
8772:
8770:
8769:
8763:
8761:
8755:
8754:
8752:
8751:
8746:
8741:
8736:
8731:
8726:
8720:
8718:
8712:
8711:
8709:
8708:
8703:
8698:
8693:
8688:
8682:
8680:
8671:
8665:
8664:
8661:
8660:
8658:
8657:
8652:
8647:
8642:
8637:
8632:
8627:
8622:
8617:
8612:
8606:
8604:
8598:
8597:
8595:
8594:
8589:
8584:
8579:
8574:
8569:
8564:
8559:
8554:
8548:
8546:
8537:
8527:
8526:
8521:
8514:
8513:
8510:
8509:
8507:
8506:
8501:
8496:
8491:
8486:
8481:
8475:
8472:
8471:
8468:
8461:
8460:
8457:
8456:
8454:
8453:
8448:
8443:
8438:
8433:
8427:
8424:
8423:
8420:
8413:
8412:
8409:
8408:
8406:
8405:
8400:
8395:
8390:
8385:
8380:
8375:
8370:
8364:
8361:
8360:
8354:
8347:
8346:
8343:
8342:
8340:
8339:
8334:
8329:
8324:
8319:
8313:
8310:
8309:
8306:
8299:
8298:
8295:
8294:
8292:
8291:
8286:
8281:
8276:
8271:
8266:
8261:
8255:
8252:
8251:
8244:
8237:
8236:
8233:
8232:
8230:
8229:
8224:
8219:
8214:
8209:
8204:
8199:
8194:
8189:
8184:
8179:
8174:
8169:
8164:
8158:
8155:
8154:
8148:
8141:
8140:
8134:
8132:
8131:
8124:
8117:
8109:
8100:
8099:
8097:
8096:
8091:
8086:
8081:
8076:
8071:
8066:
8061:
8056:
8050:
8048:
8044:
8043:
8041:
8040:
8035:
8030:
8025:
8020:
8015:
8009:
8007:
8000:
7999:
7997:
7996:
7991:
7986:
7976:
7970:
7968:
7961:
7960:
7958:
7957:
7952:
7947:
7942:
7937:
7932:
7927:
7922:
7916:
7914:
7910:
7909:
7907:
7906:
7901:
7896:
7891:
7886:
7881:
7876:
7871:
7866:
7861:
7859:Almost perfect
7856:
7850:
7848:
7844:
7843:
7841:
7840:
7835:
7830:
7825:
7820:
7815:
7810:
7805:
7800:
7795:
7790:
7785:
7780:
7774:
7772:
7768:
7767:
7760:
7758:
7756:
7755:
7750:
7745:
7740:
7735:
7730:
7724:
7722:
7718:
7717:
7714:
7712:
7711:
7704:
7697:
7689:
7683:
7682:
7654:
7648:
7642:
7639:OddPerfect.org
7636:
7628:
7609:
7604:
7598:
7578:
7577:External links
7575:
7574:
7573:
7559:
7538:
7531:
7524:
7487:
7482:
7479:
7477:
7476:
7463:
7434:
7420:
7418:
7415:
7412:
7411:
7403:
7383:
7372:Dickson, L. E.
7363:
7350:
7331:(4): 390â392.
7315:
7296:(6): 442â443.
7280:
7237:
7230:
7210:
7203:10.4171/EM/149
7196:(3): 121â126.
7176:
7157:
7144:
7091:
7044:
7041:
7037:
7031:
7027:
7007:
6958:
6911:
6876:(1): 129â146.
6855:
6826:(2): 280â286.
6806:
6747:
6696:
6677:
6658:
6627:
6584:
6549:
6518:
6505:(3): 353â359.
6485:
6449:
6382:
6339:
6306:
6255:
6209:
6158:
6131:
6080:
6054:
6019:
5960:
5934:
5923:Oddperfect.org
5915:
5887:
5876:Dickson, L. E.
5867:
5842:
5832:. Mersenne.org
5821:
5793:
5749:
5742:
5722:
5715:
5695:
5684:Dickson, L. E.
5675:
5668:
5641:
5628:
5601:
5588:
5569:
5544:
5533:Dickson, L. E.
5524:
5511:
5486:
5485:
5483:
5480:
5477:
5476:
5447:
5444:
5441:
5436:
5432:
5364:
5361:
5356:
5352:
5334:
5333:
5331:
5328:
5327:
5326:
5321:
5316:
5311:
5306:
5301:
5299:Leinster group
5296:
5289:
5286:
5255:
5164:
5155:
5146:
5137:
5124:
5111:
5102:
5093:
5084:
5073:
5070:
5069:
5068:
5061:
5054:
5038:
5033:
5028:
5025:
4999:
4994:
4979:
4967:
4964:
4959:
4955:
4931:
4928:
4925:
4920:
4916:
4912:
4907:
4904:
4901:
4897:
4877:
4876:
4875:
4861:
4860:
4859:
4847:
4844:
4841:
4837:
4833:
4830:
4827:
4823:
4819:
4816:
4813:
4809:
4805:
4802:
4799:
4795:
4791:
4788:
4785:
4781:
4777:
4774:
4771:
4767:
4763:
4752:
4740:
4737:
4732:
4728:
4725:
4722:
4716:
4711:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4683:
4678:
4675:
4670:
4665:
4662:
4657:
4652:
4649:
4644:
4639:
4636:
4631:
4626:
4623:
4618:
4613:
4610:
4605:
4600:
4597:
4592:
4587:
4584:
4555:
4552:
4549:
4546:
4543:
4540:
4535:
4531:
4511:
4504:
4479:
4476:
4463:
4462:
4461:
4460:
4444:
4441:
4438:
4435:
4432:
4427:
4423:
4419:
4415:
4410:
4406:
4403:
4383:
4380:
4377:
4374:
4371:
4366:
4362:
4358:
4355:
4352:
4342:
4325:
4318:
4311:
4305:
4298:
4292:
4278:
4273:
4270:
4267:
4264:
4261:
4258:
4255:
4251:
4247:
4244:
4241:
4238:
4235:
4213:
4206:
4200:
4182:
4176:
4158:
4152:
4142:
4136:
4126:
4113:
4106:
4100:
4099:
4098:
4097:
4082:
4079:
4076:
4073:
4068:
4065:
4062:
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:
10081:
10070:
10067:
10065:
10062:
10060:
10057:
10055:
10052:
10050:
10047:
10046:
10044:
10029:
10025:
10021:
10020:
10017:
10007:
10004:
10003:
10000:
9995:
9990:
9986:
9976:
9973:
9971:
9968:
9967:
9964:
9959:
9954:
9950:
9940:
9937:
9935:
9932:
9931:
9928:
9923:
9918:
9914:
9904:
9901:
9899:
9896:
9895:
9892:
9888:
9882:
9878:
9868:
9865:
9863:
9860:
9858:
9855:
9854:
9851:
9847:
9842:
9838:
9824:
9821:
9820:
9818:
9814:
9808:
9805:
9803:
9800:
9798:
9797:Polydivisible
9795:
9793:
9790:
9788:
9785:
9783:
9780:
9778:
9775:
9774:
9772:
9768:
9762:
9759:
9757:
9754:
9752:
9749:
9747:
9744:
9742:
9739:
9738:
9736:
9733:
9728:
9722:
9719:
9717:
9714:
9712:
9709:
9707:
9704:
9702:
9699:
9697:
9694:
9692:
9689:
9688:
9686:
9683:
9679:
9671:
9668:
9667:
9666:
9663:
9662:
9660:
9657:
9653:
9641:
9638:
9637:
9636:
9633:
9631:
9628:
9626:
9623:
9621:
9618:
9616:
9613:
9611:
9608:
9606:
9603:
9601:
9598:
9596:
9593:
9592:
9590:
9586:
9580:
9577:
9576:
9574:
9570:
9564:
9561:
9559:
9556:
9555:
9553:
9551:Digit product
9549:
9543:
9540:
9538:
9535:
9533:
9530:
9528:
9525:
9524:
9522:
9520:
9516:
9508:
9505:
9503:
9500:
9499:
9498:
9495:
9494:
9492:
9490:
9485:
9481:
9477:
9472:
9467:
9463:
9453:
9450:
9448:
9445:
9443:
9440:
9438:
9435:
9433:
9430:
9428:
9425:
9423:
9420:
9418:
9415:
9413:
9410:
9408:
9405:
9403:
9400:
9398:
9395:
9393:
9390:
9388:
9387:ErdĆsâNicolas
9385:
9383:
9380:
9378:
9375:
9374:
9371:
9366:
9362:
9356:
9352:
9338:
9335:
9333:
9330:
9329:
9327:
9325:
9321:
9315:
9312:
9310:
9307:
9305:
9302:
9300:
9297:
9296:
9294:
9292:
9288:
9282:
9279:
9277:
9274:
9272:
9269:
9267:
9264:
9262:
9259:
9257:
9254:
9253:
9251:
9249:
9245:
9239:
9236:
9234:
9231:
9230:
9228:
9226:
9222:
9216:
9213:
9211:
9208:
9206:
9205:Superabundant
9203:
9201:
9198:
9196:
9193:
9191:
9188:
9186:
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:
9107:
9105:
9103:
9099:
9095:
9091:
9087:
9082:
9078:
9068:
9065:
9063:
9060:
9058:
9055:
9053:
9050:
9048:
9045:
9043:
9040:
9038:
9035:
9033:
9030:
9028:
9025:
9023:
9020:
9018:
9015:
9013:
9010:
9009:
9006:
9002:
8997:
8993:
8983:
8980:
8978:
8975:
8973:
8970:
8968:
8965:
8964:
8961:
8957:
8952:
8948:
8938:
8935:
8933:
8930:
8928:
8925:
8923:
8920:
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:
8839:
8836:
8829:
8825:
8807:
8804:
8802:
8799:
8797:
8794:
8793:
8791:
8787:
8784:
8782:
8781:4-dimensional
8778:
8768:
8765:
8764:
8762:
8760:
8756:
8750:
8747:
8745:
8742:
8740:
8737:
8735:
8732:
8730:
8727:
8725:
8722:
8721:
8719:
8717:
8713:
8707:
8704:
8702:
8699:
8697:
8694:
8692:
8691:Centered cube
8689:
8687:
8684:
8683:
8681:
8679:
8675:
8672:
8670:
8669:3-dimensional
8666:
8656:
8653:
8651:
8648:
8646:
8643:
8641:
8638:
8636:
8633:
8631:
8628:
8626:
8623:
8621:
8618:
8616:
8613:
8611:
8608:
8607:
8605:
8603:
8599:
8593:
8590:
8588:
8585:
8583:
8580:
8578:
8575:
8573:
8570:
8568:
8565:
8563:
8560:
8558:
8555:
8553:
8550:
8549:
8547:
8545:
8541:
8538:
8536:
8535:2-dimensional
8532:
8528:
8524:
8519:
8515:
8505:
8502:
8500:
8497:
8495:
8492:
8490:
8487:
8485:
8482:
8480:
8479:Nonhypotenuse
8477:
8476:
8473:
8466:
8462:
8452:
8449:
8447:
8444:
8442:
8439:
8437:
8434:
8432:
8429:
8428:
8425:
8418:
8414:
8404:
8401:
8399:
8396:
8394:
8391:
8389:
8386:
8384:
8381:
8379:
8376:
8374:
8371:
8369:
8366:
8365:
8362:
8357:
8352:
8348:
8338:
8335:
8333:
8330:
8328:
8325:
8323:
8320:
8318:
8315:
8314:
8311:
8304:
8300:
8290:
8287:
8285:
8282:
8280:
8277:
8275:
8272:
8270:
8267:
8265:
8262:
8260:
8257:
8256:
8253:
8248:
8242:
8238:
8228:
8225:
8223:
8220:
8218:
8217:Perfect power
8215:
8213:
8210:
8208:
8207:Seventh power
8205:
8203:
8200:
8198:
8195:
8193:
8190:
8188:
8185:
8183:
8180:
8178:
8175:
8173:
8170:
8168:
8165:
8163:
8160:
8159:
8156:
8151:
8146:
8142:
8138:
8130:
8125:
8123:
8118:
8116:
8111:
8110:
8107:
8095:
8092:
8090:
8087:
8085:
8082:
8080:
8077:
8075:
8072:
8070:
8067:
8065:
8062:
8060:
8057:
8055:
8052:
8051:
8049:
8045:
8039:
8036:
8034:
8033:Polydivisible
8031:
8029:
8026:
8024:
8021:
8019:
8016:
8014:
8011:
8010:
8008:
8005:
8001:
7995:
7992:
7990:
7987:
7984:
7980:
7977:
7975:
7972:
7971:
7969:
7966:
7962:
7956:
7953:
7951:
7948:
7946:
7943:
7941:
7938:
7936:
7935:Superabundant
7933:
7931:
7928:
7926:
7923:
7921:
7918:
7917:
7915:
7911:
7905:
7904:ErdĆsâNicolas
7902:
7900:
7897:
7895:
7892:
7890:
7887:
7885:
7882:
7880:
7877:
7875:
7872:
7870:
7867:
7865:
7862:
7860:
7857:
7855:
7852:
7851:
7849:
7845:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7819:
7816:
7814:
7813:Perfect power
7811:
7809:
7806:
7804:
7801:
7799:
7796:
7794:
7791:
7789:
7786:
7784:
7781:
7779:
7776:
7775:
7773:
7769:
7764:
7754:
7751:
7749:
7746:
7744:
7741:
7739:
7736:
7734:
7731:
7729:
7726:
7725:
7723:
7719:
7710:
7705:
7703:
7698:
7696:
7691:
7690:
7687:
7673:on 2013-05-31
7672:
7668:
7664:
7660:
7655:
7652:
7649:
7646:
7643:
7640:
7637:
7635:
7629:
7624:
7623:
7618:
7615:
7610:
7608:
7605:
7603:
7600:David Moews:
7599:
7595:
7591:
7590:
7585:
7581:
7580:
7576:
7570:
7566:
7562:
7560:1-4020-2546-7
7556:
7552:
7547:
7546:
7539:
7536:
7532:
7529:
7525:
7521:
7517:
7512:
7507:
7503:
7499:
7498:
7493:
7488:
7485:
7484:
7480:
7473:
7469:
7464:
7460:
7456:
7452:
7448:
7444:
7440:
7435:
7432:
7428:
7427:
7422:
7421:
7416:
7406:
7404:9780824796969
7400:
7396:
7395:
7387:
7384:
7379:
7378:
7373:
7367:
7364:
7360:
7357:H. Novarese.
7354:
7351:
7346:
7342:
7338:
7334:
7330:
7326:
7319:
7316:
7311:
7307:
7303:
7299:
7295:
7291:
7284:
7281:
7276:
7272:
7268:
7264:
7260:
7256:
7252:
7248:
7241:
7238:
7233:
7231:9781118188613
7227:
7223:
7222:
7214:
7211:
7204:
7199:
7195:
7191:
7187:
7180:
7177:
7172:
7168:
7161:
7158:
7154:
7148:
7145:
7140:
7136:
7132:
7128:
7124:
7120:
7115:
7110:
7106:
7102:
7095:
7092:
7087:
7083:
7079:
7075:
7071:
7067:
7066:
7058:
7042:
7039:
7035:
7029:
7025:
7011:
7008:
7003:
6999:
6995:
6991:
6986:
6981:
6977:
6973:
6969:
6962:
6959:
6954:
6950:
6946:
6942:
6938:
6934:
6933:
6925:
6918:
6916:
6912:
6907:
6903:
6899:
6895:
6891:
6887:
6883:
6879:
6875:
6871:
6870:
6865:
6859:
6856:
6851:
6847:
6843:
6839:
6834:
6829:
6825:
6821:
6817:
6810:
6807:
6802:
6798:
6794:
6790:
6785:
6780:
6776:
6772:
6771:
6763:
6756:
6754:
6752:
6748:
6743:
6739:
6735:
6731:
6727:
6723:
6719:
6715:
6711:
6707:
6700:
6697:
6692:
6688:
6681:
6678:
6674:(6): 523-527.
6673:
6669:
6662:
6659:
6646:
6642:
6638:
6631:
6628:
6615:
6610:
6606:
6602:
6595:
6588:
6585:
6579:
6574:
6570:
6566:
6565:
6560:
6553:
6550:
6537:
6533:
6529:
6522:
6519:
6513:
6508:
6504:
6500:
6496:
6489:
6486:
6474:
6470:
6463:
6456:
6454:
6450:
6438:
6434:
6430:
6426:
6422:
6418:
6413:
6408:
6404:
6400:
6393:
6386:
6383:
6370:
6365:
6361:
6357:
6350:
6343:
6340:
6328:
6324:
6317:
6310:
6307:
6294:
6289:
6285:
6281:
6277:
6273:
6266:
6259:
6256:
6250:
6246:
6242:
6238:
6233:
6228:
6224:
6220:
6213:
6210:
6197:
6192:
6188:
6184:
6180:
6176:
6169:
6162:
6159:
6154:
6150:
6146:
6142:
6135:
6132:
6119:
6114:
6110:
6106:
6102:
6098:
6091:
6084:
6081:
6076:
6072:
6065:
6058:
6055:
6050:
6046:
6042:
6038:
6034:
6031:(in German).
6030:
6023:
6020:
6015:
6011:
6007:
6003:
5998:
5993:
5989:
5985:
5984:
5976:
5969:
5967:
5965:
5961:
5949:
5945:
5938:
5935:
5932:
5928:
5924:
5919:
5916:
5904:
5897:
5891:
5888:
5883:
5882:
5877:
5871:
5868:
5862:
5861:
5856:
5853:
5846:
5843:
5831:
5825:
5822:
5818:
5814:
5813:
5808:
5804:
5797:
5794:
5789:
5783:
5768:
5767:
5762:
5756:
5754:
5750:
5745:
5743:88-8358-537-2
5739:
5735:
5734:
5726:
5723:
5718:
5716:0-19-515799-0
5712:
5708:
5707:
5699:
5696:
5691:
5690:
5685:
5679:
5676:
5671:
5669:0-486-20430-8
5665:
5661:
5660:
5655:
5650:
5645:
5642:
5638:
5632:
5629:
5624:
5617:
5616:
5613:
5605:
5602:
5598:
5592:
5589:
5585:
5581:
5580:
5573:
5570:
5558:
5554:
5548:
5545:
5540:
5539:
5534:
5528:
5525:
5521:
5515:
5512:
5501:
5497:
5491:
5488:
5481:
5472:
5468:
5445:
5442:
5439:
5434:
5430:
5419:
5411:
5403:
5397:
5384:
5380:
5362:
5359:
5354:
5350:
5339:
5336:
5329:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5307:
5305:
5302:
5300:
5297:
5295:
5292:
5291:
5287:
5285:
5283:
5282:weird numbers
5278:
5273:
5271:
5243:
5238:
5234:
5230:
5226:
5222:
5218:
5214:
5209:
5207:
5203:
5199:
5195:
5191:
5187:
5183:
5179:
5171:
5162:
5153:
5144:
5135:
5131:
5122:
5118:
5117:Superabundant
5109:
5100:
5091:
5082:
5081:Euler diagram
5078:
5071:
5066:
5062:
5059:
5055:
5052:
5031:
5023:
5015:
4997:
4992:
4985:is less than
4984:
4980:
4965:
4962:
4957:
4953:
4945:
4926:
4923:
4918:
4914:
4905:
4902:
4899:
4895:
4886:
4882:
4878:
4873:
4869:
4868:
4866:
4862:
4845:
4842:
4839:
4835:
4831:
4828:
4825:
4821:
4817:
4814:
4811:
4807:
4803:
4800:
4797:
4793:
4789:
4786:
4783:
4779:
4775:
4772:
4769:
4765:
4761:
4753:
4738:
4735:
4730:
4726:
4723:
4720:
4714:
4709:
4705:
4702:
4699:
4696:
4693:
4690:
4687:
4681:
4676:
4673:
4668:
4663:
4660:
4655:
4650:
4647:
4642:
4637:
4634:
4629:
4624:
4621:
4616:
4611:
4608:
4603:
4598:
4595:
4590:
4585:
4582:
4572:
4571:
4569:
4553:
4550:
4547:
4541:
4533:
4529:
4520:
4516:
4512:
4509:
4508:Gallardo 2010
4505:
4502:
4501:Makowski 1962
4498:
4494:
4493:
4492:
4490:
4486:
4478:Minor results
4477:
4474:
4470:
4468:
4442:
4439:
4436:
4433:
4430:
4425:
4421:
4417:
4413:
4408:
4404:
4401:
4381:
4378:
4375:
4372:
4369:
4364:
4360:
4356:
4353:
4350:
4343:
4340:
4337:
4336:
4333:
4328:
4324:
4317:
4312:
4308:
4304:
4297:
4293:
4276:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4249:
4242:
4239:
4236:
4225:
4221:
4216:
4212:
4205:
4201:
4198:
4194:
4190:
4185:
4181:
4177:
4174:
4170:
4166:
4161:
4157:
4153:
4150:
4145:
4141:
4137:
4134:
4129:
4125:
4121:
4120:
4119:
4116:
4112:
4105:
4080:
4077:
4074:
4071:
4066:
4063:
4060:
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:
9761:Transposable
9625:Narcissistic
9532:Digital root
9452:Super-Poulet
9412:JordanâPĂłlya
9361:prime factor
9303:
9266:Noncototient
9233:Almost prime
9215:Superperfect
9190:Refactorable
9185:Quasiperfect
9169:
9160:Hyperperfect
9001:Pseudoprimes
8972:WallâSunâSun
8907:Ordered Bell
8877:FussâCatalan
8789:non-centered
8739:Dodecahedral
8716:non-centered
8602:non-centered
8504:Wolstenholme
8249:× 2 ± 1
8246:
8245:Of the form
8212:Eighth power
8192:Fourth power
8094:Superperfect
8089:Refactorable
7884:Superperfect
7879:Hyperperfect
7864:Quasiperfect
7853:
7748:Prime factor
7675:. Retrieved
7671:the original
7662:
7620:
7587:
7544:
7534:
7527:
7501:
7495:
7471:
7467:
7442:
7438:
7424:
7393:
7386:
7376:
7366:
7358:
7353:
7328:
7324:
7318:
7293:
7289:
7283:
7250:
7246:
7240:
7220:
7213:
7193:
7189:
7179:
7170:
7166:
7160:
7152:
7147:
7107:(1): 15â21.
7104:
7100:
7094:
7072:(1): 25â28.
7069:
7063:
7010:
6978:(1): 13â15.
6975:
6971:
6961:
6939:(1): 70â76.
6936:
6930:
6873:
6867:
6858:
6823:
6819:
6809:
6774:
6768:
6712:(1): 52â53.
6709:
6705:
6699:
6690:
6686:
6680:
6671:
6667:
6661:
6649:. Retrieved
6644:
6640:
6630:
6618:. Retrieved
6604:
6600:
6587:
6568:
6562:
6552:
6540:. Retrieved
6535:
6531:
6521:
6502:
6498:
6488:
6476:. Retrieved
6472:
6468:
6440:. Retrieved
6412:math/0602485
6402:
6398:
6385:
6373:. Retrieved
6359:
6355:
6342:
6330:. Retrieved
6326:
6322:
6309:
6297:. Retrieved
6275:
6271:
6258:
6222:
6218:
6212:
6200:. Retrieved
6178:
6174:
6161:
6144:
6140:
6134:
6122:. Retrieved
6100:
6096:
6083:
6074:
6070:
6057:
6032:
6028:
6022:
5987:
5981:
5953:10 September
5951:. Retrieved
5947:
5937:
5918:
5906:. Retrieved
5902:
5890:
5880:
5870:
5858:
5845:
5834:. Retrieved
5830:"GIMPS Home"
5824:
5810:
5796:
5770:. Retrieved
5764:
5732:
5725:
5705:
5698:
5688:
5678:
5658:
5644:
5636:
5631:
5622:
5615:
5611:
5604:
5591:
5577:
5572:
5560:. Retrieved
5556:
5547:
5537:
5527:
5514:
5503:. Retrieved
5499:
5490:
5417:
5409:
5401:
5382:
5338:
5274:
5236:
5232:
5228:
5224:
5220:
5210:
5189:
5175:
5151:
5058:digital root
5013:
4982:
4944:Fermat prime
4864:
4567:
4518:
4496:
4481:
4472:
4464:
4338:
4331:
4326:
4322:
4315:
4306:
4302:
4295:
4223:
4219:
4214:
4210:
4203:
4196:
4192:
4188:
4183:
4179:
4172:
4159:
4155:
4143:
4139:
4127:
4123:
4114:
4110:
4103:
4101:
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:
9782:Extravagant
9777:Equidigital
9732:permutation
9691:Palindromic
9665:Automorphic
9563:Sum-product
9542:Sum-product
9497:Persistence
9392:ErdĆsâWoods
9314:Untouchable
9195:Semiperfect
9145:Hemiperfect
8806:Tesseractic
8744:Icosahedral
8724:Tetrahedral
8655:Dodecagonal
8356:Recursively
8227:Prime power
8202:Sixth power
8197:Fifth power
8177:Power of 10
8135:Classes of
8018:Extravagant
8013:Equidigital
7974:Untouchable
7894:Semiperfect
7874:Hemiperfect
7803:Square-free
7667:Brady Haran
7663:Numberphile
7533:Riesel, H.
6620:29 November
6035:: 202â211.
5903:Harvard.edu
5213:fixed point
5176:The sum of
5065:square-free
4515:reciprocals
4485:Richard Guy
4226: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
10043:Categories
9994:Graphemics
9867:Pernicious
9721:Undulating
9696:Pandigital
9670:Trimorphic
9271:Nontotient
9120:Arithmetic
8734:Octahedral
8635:Heptagonal
8625:Pentagonal
8610:Triangular
8451:SierpiĆski
8373:Jacobsthal
8172:Power of 3
8167:Power of 2
8054:Arithmetic
8047:Other sets
8006:-dependent
7677:2013-04-02
7569:1079.11001
7290:Arch. Math
7190:Elem. Math
7167:Elem. Math
7114:1706.09341
6693:: 896â904.
6651:7 December
6614:2303.11974
6332:6 December
6232:1810.11734
6014:1263.11005
5836:2022-07-21
5505:2024-03-21
5482:References
5398:âthat is,
5240:, and the
5194:numerology
2702:1 + 9 = 10
1535:) and the
1133:Nicomachus
586:is prime (
473:See also:
343:Nicomachus
18:8589869056
9751:Parasitic
9600:Factorion
9527:Digit sum
9519:Digit sum
9337:Fortunate
9324:Primorial
9238:Semiprime
9175:Practical
9140:Descartes
9135:Deficient
9125:Betrothed
8967:Wieferich
8796:Pentatope
8759:pyramidal
8650:Decagonal
8645:Nonagonal
8640:Octagonal
8630:Hexagonal
8489:Practical
8436:Congruent
8368:Fibonacci
8332:Loeschian
8084:Descartes
8059:Deficient
7994:Betrothed
7899:Practical
7788:Semiprime
7783:Composite
7622:MathWorld
7594:EMS Press
7459:115983363
7345:122353640
7325:Math. Ann
7310:122525522
7275:125545112
7173:(5): 109.
7139:119175632
7131:1730-6302
7078:0015-0517
7043:β
7030:α
6994:1088-6826
6945:0015-0517
6906:122452828
6890:1435-5345
6842:1446-8107
6793:0025-5718
6742:121251041
6726:1420-8938
6538:: A14âA22
6375:13 August
6077:(4): 244.
6049:120754476
6006:0025-5718
5860:MathWorld
5440:−
5360:−
5182:deficient
5170:Deficient
5161:Composite
5063:The only
4903:−
4724:⋅
4530:σ
4467:Sylvester
4465:In 1888,
4354:≤
4277:α
4263:≤
4257:≤
4240:−
4222:ones and
4167:3) or 2 (
4078:
4064:
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
9823:Friedman
9756:Primeval
9701:Repdigit
9658:-related
9605:Kaprekar
9579:Meertens
9502:Additive
9489:dynamics
9397:Friendly
9309:Sociable
9299:Amicable
9110:Abundant
9090:dynamics
8912:Schröder
8902:Narayana
8872:Eulerian
8862:Delannoy
8857:Dedekind
8678:centered
8544:centered
8431:Amenable
8388:Narayana
8378:Leonardo
8274:Mersenne
8222:Powerful
8162:Achilles
8069:Solitary
8064:Friendly
7989:Sociable
7979:Amicable
7967:-related
7920:Abundant
7818:Achilles
7808:Powerful
7721:Overview
7474:: 69â72.
7426:Elements
7423:Euclid,
7374:(1919).
6601:Integers
6542:23 March
6532:Integers
6478:7 August
6469:Integers
6442:30 March
6323:Integers
6299:30 March
6249:62885986
6202:30 March
6124:30 March
5927:Archived
5878:(1919).
5782:cite web
5686:(1919).
5656:(1925).
5535:(1919).
5500:oeis.org
5288:See also
5202:sociable
5198:amicable
5186:abundant
5090:Abundant
5049:, using
5012:, where
4469:stated:
4321:= ... =
4138:Not all
4122:Not all
3350:> 10.
2280:33550336
589:Elements
170:Elements
59:divisors
9996:related
9960:related
9924:related
9922:Sorting
9807:Vampire
9792:Harshad
9734:related
9706:Repunit
9620:Lychrel
9595:Dudeney
9447:StĂžrmer
9442:Sphenic
9427:Regular
9365:divisor
9304:Perfect
9200:Sublime
9170:Perfect
8897:Motzkin
8852:Catalan
8393:Padovan
8327:Leyland
8322:Idoneal
8317:Hilbert
8289:Woodall
8074:Sublime
8028:Harshad
7854:Perfect
7838:Unusual
7828:Regular
7798:Sphenic
7733:Divisor
7647:(GIMPS)
7596:, 2001
7520:2005530
7417:Sources
7267:3619053
7086:0354538
7002:0292740
6953:0786364
6898:0044579
6850:0869751
6801:2904601
6734:0258723
6647:: 23â30
6437:2767519
6417:Bibcode
6280:Bibcode
6183:Bibcode
6105:Bibcode
5908:16 June
5772:28 July
5471:A002515
5469::
5215:of the
5190:perfect
5168:
5159:
5152:Perfect
5150:
5141:
5128:
5115:
5106:
5097:
5088:
4335:, then
4154:If all
3618:,
3612:where:
3503:, then
1373:in the
1370:A000043
333:History
181:perfect
159:is the
9862:Odious
9787:Frugal
9741:Cyclic
9730:Digit-
9437:Smooth
9422:Pronic
9382:Cyclic
9359:Other
9332:Euclid
8982:Wilson
8956:Primes
8615:Square
8484:Polite
8446:Riesel
8441:Knödel
8403:Perrin
8284:Thabit
8269:Fermat
8259:Cullen
8182:Square
8150:Powers
8023:Frugal
7983:Triple
7823:Smooth
7793:Pronic
7567:
7557:
7518:
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7401:
7343:
7308:
7273:
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5139:
5126:
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5095:
5086:
4858:, etc.
2704:, and
2658:= 8128
1194:where
1131:While
496:Euclid
447:Origen
404:where
193:Euclid
132:where
9903:Prime
9898:Lucky
9887:sieve
9816:Other
9802:Smith
9682:Digit
9640:Happy
9615:Keith
9588:Other
9432:Rough
9402:Giuga
8867:Euler
8729:Cubic
8383:Lucas
8279:Proth
8038:Smith
7955:Weird
7833:Rough
7778:Prime
7553:â98.
7516:JSTOR
7455:S2CID
7341:S2CID
7306:S2CID
7271:S2CID
7263:JSTOR
7135:S2CID
7109:arXiv
7060:(PDF)
6927:(PDF)
6902:S2CID
6765:(PDF)
6738:S2CID
6609:arXiv
6597:(PDF)
6465:(PDF)
6433:S2CID
6407:arXiv
6395:(PDF)
6352:(PDF)
6319:(PDF)
6268:(PDF)
6245:S2CID
6227:arXiv
6171:(PDF)
6093:(PDF)
6067:(PDF)
6045:S2CID
5978:(PDF)
5899:(PDF)
5619:(PDF)
5562:9 May
5394:is a
5330:Notes
5143:Weird
5132:and
5119:and
3287:(cf.
3030:11100
2682:= 903
2650:= 496
1303:GIMPS
950:8128.
653:with
347:every
187:, or
185:ideal
53:is a
9857:Evil
9537:Self
9487:and
9377:Blum
9088:and
8892:Lobb
8847:Cake
8842:Bell
8592:Star
8499:Ulam
8398:Pell
8187:Cube
8004:Base
7631:OEIS
7555:ISBN
7472:1937
7443:1941
7399:ISBN
7226:ISBN
7127:ISSN
7074:ISSN
6990:ISSN
6941:ISSN
6886:ISSN
6838:ISSN
6789:ISSN
6722:ISSN
6653:2018
6622:2023
6544:2021
6480:2021
6444:2011
6377:2015
6334:2021
6301:2011
6204:2011
6126:2011
6002:ISSN
5955:2020
5910:2023
5788:link
5774:2024
5738:ISBN
5711:ISBN
5664:ISBN
5564:2018
5467:OEIS
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5227:) =
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4055:>
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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
9975:Ban
9363:or
8882:Lah
7565:Zbl
7506:doi
7447:doi
7333:doi
7329:131
7298:doi
7255:doi
7198:doi
7119:doi
7105:156
6980:doi
6878:doi
6874:188
6828:doi
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2960:110
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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:)
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