3014:
45:
5275:
2712:
273:
in 1972. Fermat and
Descartes also rediscovered pairs of amicable numbers known to Arab mathematicians. Euler also discovered dozens of new pairs. The second smallest pair, (1184, 1210), was discovered in 1867 by 16-year-old B. Nicolò I. Paganini (not to be confused with the composer and violinist),
1251:
or both odd. It is not known whether an even-odd pair of amicable numbers exists, but if it does, the even number must either be a square number or twice one, and the odd number must be a square number. However, amicable numbers where the two members have different smallest prime factors do exist:
931:
599:
710:
1804:
1696:
422:
In particular, the two rules below produce only even amicable pairs, so they are of no interest for the open problem of finding amicable pairs coprime to 210 = 2·3·5·7, while over 1000 pairs coprime to 30 = 2·3·5 are known .
451:
130:). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.
1305:). Although all amicable pairs up to 10,000 are even pairs, the proportion of odd amicable pairs increases steadily towards higher numbers, and presumably there are more of them than of even amicable pairs (
133:
The first ten amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), and (66928, 66992). (sequence
715:
456:
2769:
1562:
1497:
2295:
1756:
Sociable numbers are the numbers in cyclic lists of numbers (with a length greater than 2) where each number is the sum of the proper divisors of the preceding number. For example,
3377:
182:
proper divisors, in other words a number which forms an aliquot sequence of period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as
1441:
1400:
2497:
2448:
1759:
1959:
1853:
1359:
1902:
1570:
2181:
1922:
1873:
1995:
681:. In order for Ibn Qurrah's formula to produce an amicable pair, two consecutive Thabit numbers must be prime; this severely restricts the possible values of
2732:
926:{\displaystyle {\begin{aligned}p&=(2^{n-m}+1)\times 2^{m}-1,\\q&=(2^{n-m}+1)\times 2^{n}-1,\\r&=(2^{n-m}+1)^{2}\times 2^{m+n}-1,\end{aligned}}}
1264:
of the two must be greater than 10. Also, a pair of coprime amicable numbers cannot be generated by Thabit's formula (above), nor by any similar formula.
2727:
2025:
2957:
1739:
1722:
1712:
1302:
1285:
1236:
1098:
155:
145:
140:
3370:
204:
214:, who credited them with many mystical properties. A general formula by which some of these numbers could be derived was invented circa 850 by the
2750:
2349:
2222:
2674:
Patrick
Costello, Ranthony A. C. Edmonds. "Gaussian Amicable Pairs." Missouri Journal of Mathematical Sciences, 30(2) 107-116 November 2018.
2275:
997:
with no others being known. Euler (1747 & 1750) overall found 58 new pairs increasing the number of pairs that were then known to 61.
4177:
3363:
2214:
419:
While these rules do generate some pairs of amicable numbers, many other pairs are known, so these rules are by no means comprehensive.
1278:
noted that most even amicable pairs known at his time have sums divisible by 9, and a rule for characterizing the exceptions (sequence
4172:
4187:
4167:
235:
3003:
2289:
2016:
2372:
4880:
4460:
3200:
3013:
4182:
2365:
4966:
2860:
594:{\displaystyle {\begin{aligned}p&=3\times 2^{n-1}-1,\\q&=3\times 2^{n}-1,\\r&=9\times 2^{2n-1}-1,\end{aligned}}}
2950:
2281:
251:
227:
4282:
1295:
numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by ten approaches 100% (sequence
5299:
4632:
3951:
3744:
4667:
4637:
4312:
4302:
3154:
2158:
4808:
4222:
3956:
3936:
2543:
4498:
2239:
4662:
696:. The second group of lemmas deals more specifically with the formation of perfect, abundant and deficient numbers.
5309:
4757:
4380:
4137:
3946:
3928:
3822:
3812:
3802:
3190:
2605:
2392:
4642:
2245:(in Italian). Universita degli Studi di Firenze: Dipartimento di Sistemi e Informatica. p. 59. Archived from
5304:
4885:
4430:
4051:
3837:
3832:
3827:
3817:
3794:
3175:
1502:
3870:
2943:
2092:
1446:
4127:
4996:
4961:
4747:
4657:
4531:
4506:
4415:
4405:
4017:
3999:
3919:
3329:
3195:
3119:
2054:
2038:
contains a story ('Mathematical
Aphrodisiac' by Alex Galt) in which amicable numbers play an important role.
1048:
5256:
4526:
4400:
4031:
3807:
3587:
3514:
3180:
3139:
2076:
4511:
4365:
4292:
3447:
3109:
2978:
2491:
2442:
1261:
231:
5220:
4860:
692:
divided into two groups. The first three lemmas deal with the determination of the aliquot parts of a
44:
5153:
5047:
5011:
4752:
4475:
4455:
4272:
3941:
3729:
3701:
3283:
3185:
2573:
2104:
1925:
1248:
442:
243:
1405:
1364:
4875:
4739:
4734:
4702:
4465:
4440:
4435:
4410:
4340:
4336:
4267:
4157:
3989:
3785:
3754:
3344:
3339:
3134:
3129:
3114:
3053:
2462:
Hagis, Peter, Jr. (1970). "Lower bounds for relatively prime amicable numbers of opposite parity".
2084:
1076:
689:
219:
5274:
5278:
5032:
5027:
4941:
4915:
4813:
4792:
4564:
4445:
4395:
4317:
4287:
4227:
3994:
3974:
3905:
3618:
3268:
3263:
3224:
3144:
3124:
2896:
2657:
2597:
2535:
1799:{\displaystyle 1264460\mapsto 1547860\mapsto 1727636\mapsto 1305184\mapsto 1264460\mapsto \dots }
4162:
246:(16th century) discovered the pair (9363584, 9437056), though this has often been attributed to
5172:
5117:
4971:
4946:
4920:
4697:
4375:
4370:
4297:
4277:
4262:
3984:
3966:
3885:
3875:
3860:
3638:
3623:
3304:
3244:
2784:
2746:
2649:
2589:
2345:
2285:
2218:
2130:
2112:
262:
247:
171:
1931:
1825:
5208:
5001:
4587:
4559:
4541:
4425:
4390:
4385:
4352:
4046:
4009:
3900:
3895:
3890:
3880:
3852:
3739:
3691:
3686:
3643:
3582:
3334:
3309:
3215:
3149:
3033:
2993:
2756:
2689:
2639:
2581:
2527:
2471:
2422:
2173:
2067:
1962:
1815:
1691:{\displaystyle \sigma (n_{1})=\sigma (n_{2})=\dots =\sigma (n_{k})=n_{1}+n_{2}+\dots +n_{k}}
258:
167:
116:
38:
2483:
2434:
1332:
5184:
5073:
5006:
4932:
4855:
4829:
4647:
4360:
4217:
4152:
4122:
4112:
4107:
3773:
3681:
3628:
3472:
3412:
3319:
3314:
3239:
3233:
3170:
3068:
3058:
2988:
2760:
2479:
2430:
2376:
2136:
1998:
1878:
1751:
1702:
1252:
there are seven such pairs known. Also, every known pair shares at least one common prime
693:
183:
2804:
2344:. Vol. 156. Dordrecht, Boston, London: Kluwer Academic Publishers. p. 278,279.
2912:
2577:
2369:
5189:
5057:
5042:
4906:
4870:
4845:
4721:
4692:
4677:
4554:
4450:
4420:
4147:
4102:
3979:
3577:
3572:
3567:
3539:
3524:
3437:
3422:
3400:
3387:
3324:
3278:
3104:
3088:
3078:
3048:
2913:"Amicable Numbers (Formal proof development in Isabelle/HOL, Archive of Formal Proofs)"
2585:
2271:
2002:
1907:
1858:
1819:
1275:
1271:
showed that the density of amicable numbers, relative to the positive integers, was 0.
266:
211:
175:
64:
56:
2935:
1968:
5293:
5112:
5096:
5037:
4991:
4687:
4672:
4582:
4307:
3865:
3734:
3696:
3653:
3534:
3519:
3509:
3467:
3457:
3432:
3273:
3073:
3063:
3043:
2775:
2723:
2718:
2539:
2046:
2021:
1268:
678:
439:
5148:
5137:
5052:
4890:
4865:
4782:
4682:
4652:
4627:
4611:
4516:
4483:
4232:
4206:
4117:
4056:
3633:
3529:
3462:
3442:
3417:
3288:
3205:
3083:
3028:
2998:
2117:
974:
are a pair of amicable numbers. Thābit ibn Qurra's theorem corresponds to the case
953:
619:
270:
215:
127:
123:
2787:
2177:
2083:
Amicable numbers (220, 284) are referenced in episode 13 of the 2017 Korean drama
31:
2872:
5107:
4982:
4787:
4251:
4142:
4097:
4092:
3842:
3749:
3648:
3477:
3452:
3427:
2868:
2821:
2059:
2031:
435:
is a method for discovering amicable numbers invented in the 9th century by the
1715:), and (3270960, 3361680, 3461040, 3834000) is an amicable quadruple (sequence
5244:
5225:
4521:
4132:
2888:
2844:
2693:
2531:
3355:
2653:
2593:
48:
Demonstration, with rods, of the amicability of the pair of numbers (220,284)
4850:
4777:
4769:
4574:
4488:
3606:
3038:
2826:
2809:
2792:
2512:
2246:
4951:
1729:
60:
2601:
2561:
2388:
1291:
According to the sum of amicable pairs conjecture, as the number of the
163:) It is unknown if there are infinitely many pairs of amicable numbers.
17:
4956:
4615:
2983:
2661:
1257:
1253:
1068:
945:
707:
is a generalization of the Thâbit ibn Qurra theorem. It states that if
611:
193:
2413:
Hagis, Peter, Jr. (1969). "On relatively prime odd amicable numbers".
2342:
The development of Arabic mathematics: between arithmetic and algebra
1732:
are defined analogously and generalizes this a bit further (sequence
2644:
2627:
2475:
2426:
3254:
2717:
This article incorporates text from a publication now in the
43:
2240:"Introduzione alla matematica: La matematica della scuola media"
436:
239:
223:
5242:
5206:
5170:
5134:
5094:
4719:
4608:
4334:
4249:
4204:
4081:
3771:
3718:
3670:
3604:
3556:
3494:
3398:
3359:
2939:
265:(1596–1650), to whom it is sometimes ascribed, and extended by
640:
are a pair of amicable numbers. This formula gives the pairs
2628:"On Divisibility by Nine of the Sums of Even Amicable Pairs"
1734:
1717:
1707:
1306:
1297:
1292:
1280:
1247:
In every known case, the numbers of a pair are either both
1231:
1093:
159:
149:
135:
2771:
The
Penguin Dictionary of Curious and Interesting Numbers
673:, but no other such pairs are known. Numbers of the form
2843:
M. García; J.M. Pedersen; H.J.J. te Riele (2003-07-31).
2686:
Distributed cycle detection in large-scale sparse graphs
2889:"MegaFavNumbers - The Even Amicable Numbers Conjecture"
688:
To establish the theorem, Thâbit ibn Qurra proved nine
2921:
2688:, Simpósio Brasileiro de Pesquisa Operacional (SBPO),
2317:
1971:
1934:
1910:
1881:
1861:
1828:
1762:
1573:
1505:
1449:
1408:
1367:
1335:
985:. Euler's rule creates additional amicable pairs for
713:
454:
2684:
Rocha, Rodrigo
Caetano; Thatte, Bhalchandra (2015),
5066:
5020:
4980:
4931:
4905:
4838:
4822:
4801:
4768:
4733:
4573:
4540:
4497:
4474:
4351:
4039:
4030:
4008:
3965:
3927:
3918:
3851:
3793:
3784:
3297:
3253:
3214:
3163:
3097:
3021:
2971:
2159:"New Amicable Pairs Of Type (2; 2) And Type (3; 2)"
2052:Amicable numbers are mentioned in the French novel
2041:Amicable numbers are featured briefly in the novel
411:There are over 1,000,000,000 known amicable pairs.
30:"Amicable" redirects here. For the definition, see
2074:Amicable numbers are featured in the visual novel
1989:
1953:
1916:
1896:
1867:
1847:
1798:
1690:
1556:
1491:
1435:
1394:
1353:
925:
593:
274:having been overlooked by earlier mathematicians.
2110:Amicable numbers are mentioned in the 2020 novel
2090:Amicable numbers are featured in the Greek movie
1260:amicable numbers exists, though if any does, the
1499:. This can be generalized to larger tuples, say
226:mathematicians who studied amicable numbers are
2911:Koutsoukou-Argyraki, Angeliki (4 August 2020).
1229:belonging to any other amicable pair (sequence
257:Thābit ibn Qurra's formula was rediscovered by
67:of each is equal to the other number. That is,
2745:. Dordrecht: Kluwer Academic. pp. 32–36.
3371:
2951:
2139:- Three-number variation of Amicable numbers.
1997:. Two special cases are loops that represent
1310:
8:
2736:(11th ed.). Cambridge University Press.
2496:: CS1 maint: multiple names: authors list (
2447:: CS1 maint: multiple names: authors list (
107:is equal to the sum of positive divisors of
2099:Amicable numbers are discussed in the book
2065:Amicable numbers are mentioned in the JRPG
2014:Amicable numbers are featured in the novel
1557:{\displaystyle (n_{1},n_{2},\ldots ,n_{k})}
201:Are there infinitely many amicable numbers?
178:, which is a number that equals the sum of
5239:
5203:
5167:
5131:
5091:
4765:
4730:
4716:
4605:
4348:
4331:
4246:
4201:
4078:
4036:
3924:
3790:
3781:
3768:
3715:
3672:Possessing a specific set of other numbers
3667:
3601:
3553:
3491:
3395:
3378:
3364:
3356:
2958:
2944:
2936:
2741:Sándor, Jozsef; Crstici, Borislav (2004).
2204:
2202:
1904:denotes the sum of the proper divisors of
276:
166:A pair of amicable numbers constitutes an
122:The smallest pair of amicable numbers is (
27:Pair of integers related by their divisors
2643:
1970:
1939:
1933:
1909:
1880:
1860:
1833:
1827:
1761:
1682:
1663:
1650:
1634:
1606:
1584:
1572:
1545:
1526:
1513:
1504:
1492:{\displaystyle \sigma (m)=\sigma (n)=m+n}
1448:
1407:
1366:
1334:
1221:is twin if there are no integers between
898:
885:
863:
827:
799:
763:
735:
714:
712:
563:
524:
479:
455:
453:
2932:(database of all known amicable numbers)
2370:An Evening with Leonhard Euler – YouTube
2335:
2333:
2034:'s collection of short stories entitled
2001:and cycles of length two that represent
269:(1707–1783). It was extended further by
2149:
205:(more unsolved problems in mathematics)
2489:
2440:
2238:Sprugnoli, Renzo (27 September 2005).
1701:For example, (1980, 2016, 2556) is an
1013:) be a pair of amicable numbers with
7:
1256:. It is not known whether a pair of
2966:Divisibility-based sets of integers
2520:Publicationes Mathematicae Debrecen
2215:Mathematical Association of America
210:Amicable numbers were known to the
2586:10.1038/scientificamerican0368-121
174:2. A related concept is that of a
25:
3004:Fundamental theorem of arithmetic
2017:The Housekeeper and the Professor
1806:are sociable numbers of order 4.
1443:which can be written together as
1175:, the greatest common divisor is
1133:prime factors respectively, then
254:in this area has been forgotten.
5273:
4881:Perfect digit-to-digit invariant
3012:
2861:"220 and 284 (Amicable Numbers)"
2710:
2549:from the original on 2022-10-09.
2157:Costello, Patrick (1 May 2002).
2899:from the original on 2021-11-23
2608:from the original on 2022-09-25
2395:from the original on 2021-07-18
2298:from the original on 2023-09-12
2187:from the original on 2008-02-29
1316:Gaussian amicable pairs exist.
196:Unsolved problem in mathematics
59:related in such a way that the
1984:
1972:
1891:
1885:
1810:Searching for sociable numbers
1790:
1784:
1778:
1772:
1766:
1640:
1627:
1612:
1599:
1590:
1577:
1551:
1506:
1474:
1468:
1459:
1453:
1436:{\displaystyle \sigma (n)-n=m}
1418:
1412:
1395:{\displaystyle \sigma (m)-m=n}
1377:
1371:
1348:
1336:
882:
856:
817:
792:
753:
728:
1:
3720:Expressible via specific sums
2282:American Mathematical Society
2178:10.1090/S0025-5718-02-01414-X
2009:References in popular culture
278:The first ten amicable pairs
2743:Handbook of number theory II
2209:Sandifer, C. Edward (2007).
4809:Multiplicative digital root
2036:True Tales of American Life
1101:); otherwise, it is called
5326:
2845:"Amicable pairs, a survey"
2632:Mathematics of Computation
2464:Mathematics of Computation
2415:Mathematics of Computation
2166:Mathematics of Computation
1749:
36:
29:
5269:
5252:
5238:
5216:
5202:
5180:
5166:
5144:
5130:
5103:
5090:
4886:Perfect digital invariant
4729:
4715:
4623:
4604:
4461:Superior highly composite
4347:
4330:
4258:
4245:
4213:
4200:
4088:
4077:
3780:
3767:
3725:
3714:
3677:
3666:
3614:
3600:
3563:
3552:
3505:
3490:
3408:
3394:
3201:Superior highly composite
3010:
2694:10.13140/RG.2.1.1233.8640
2532:10.5486/PMD.1955.4.1-2.16
433:Thābit ibn Qurrah theorem
427:Thābit ibn Qurrah theorem
4499:Euler's totient function
4283:Euler–Jacobi pseudoprime
3558:Other polynomial numbers
3098:Constrained divisor sums
2805:"Thâbit ibn Kurrah Rule"
2560:Gardner, Martin (1968).
2133:(quasi-amicable numbers)
2093:The Other Me (2016 film)
1818:can be represented as a
37:Not to be confused with
4313:Somer–Lucas pseudoprime
4303:Lucas–Carmichael number
4138:Lazy caterer's sequence
2733:Encyclopædia Britannica
2340:Rashed, Roshdi (1994).
2277:Mathematical Magic Show
1954:{\displaystyle G_{n,s}}
1848:{\displaystyle G_{n,s}}
1049:greatest common divisor
4188:Wedderburn–Etherington
3588:Lucky numbers of Euler
1991:
1955:
1918:
1898:
1869:
1855:, for a given integer
1849:
1800:
1692:
1558:
1493:
1437:
1396:
1355:
927:
595:
252:Eastern mathematicians
250:. Much of the work of
49:
4476:Prime omega functions
4293:Frobenius pseudoprime
4083:Combinatorial numbers
3952:Centered dodecahedral
3745:Primary pseudoperfect
2979:Integer factorization
2922:"Amicable pairs list"
2513:"On amicable numbers"
2389:"Amicable pairs news"
2318:"Amicable pairs list"
1992:
1956:
1919:
1899:
1870:
1850:
1801:
1693:
1559:
1494:
1438:
1397:
1356:
1354:{\displaystyle (m,n)}
928:
596:
47:
4935:-composition related
4735:Arithmetic functions
4337:Arithmetic functions
4273:Elliptic pseudoprime
3957:Centered icosahedral
3937:Centered tetrahedral
2562:"Mathematical Games"
2511:Erdős, Paul (2022).
2252:on 13 September 2012
2055:The Parrot's Theorem
1969:
1965:within the interval
1932:
1908:
1897:{\displaystyle s(k)}
1879:
1859:
1826:
1760:
1571:
1503:
1447:
1406:
1365:
1333:
711:
699:
452:
415:Rules for generation
244:Muhammad Baqir Yazdi
5300:Arithmetic dynamics
4861:Kaprekar's constant
4381:Colossally abundant
4268:Catalan pseudoprime
4168:Schröder–Hipparchus
3947:Centered octahedral
3823:Centered heptagonal
3813:Centered pentagonal
3803:Centered triangular
3403:and related numbers
3191:Colossally abundant
3022:Factorization forms
2820:Weisstein, Eric W.
2803:Weisstein, Eric W.
2778:. pp. 145–147.
2626:Lee, Elvin (1969).
2578:1968SciAm.218c.121G
2566:Scientific American
1564:, where we require
1205:Twin amicable pairs
1197:is regular of type
279:
32:Wiktionary:amicable
5279:Mathematics portal
5221:Aronson's sequence
4967:Smarandache–Wellin
4724:-dependent numbers
4431:Primitive abundant
4318:Strong pseudoprime
4308:Perrin pseudoprime
4288:Fermat pseudoprime
4228:Wolstenholme prime
4052:Squared triangular
3838:Centered decagonal
3833:Centered nonagonal
3828:Centered octagonal
3818:Centered hexagonal
3176:Primitive abundant
3164:With many divisors
2920:Chernykh, Sergei.
2785:Weisstein, Eric W.
2768:Wells, D. (1987).
2375:2016-05-16 at the
2316:Chernykh, Sergei.
2217:. pp. 49–55.
2043:The Stranger House
1987:
1951:
1914:
1894:
1865:
1845:
1796:
1688:
1554:
1489:
1433:
1392:
1351:
1163:For example, with
995:) = (1,8), (29,40)
923:
921:
664:(9363584, 9437056)
591:
589:
448:It states that if
277:
55:are two different
50:
5310:Integer sequences
5287:
5286:
5265:
5264:
5234:
5233:
5198:
5197:
5162:
5161:
5126:
5125:
5086:
5085:
5082:
5081:
4901:
4900:
4711:
4710:
4600:
4599:
4596:
4595:
4542:Aliquot sequences
4353:Divisor functions
4326:
4325:
4298:Lucas pseudoprime
4278:Euler pseudoprime
4263:Carmichael number
4241:
4240:
4196:
4195:
4073:
4072:
4069:
4068:
4065:
4064:
4026:
4025:
3914:
3913:
3871:Square triangular
3763:
3762:
3710:
3709:
3662:
3661:
3596:
3595:
3548:
3547:
3486:
3485:
3353:
3352:
2752:978-1-4020-2546-4
2351:978-0-7923-2565-9
2224:978-0-88385-563-8
2131:Betrothed numbers
2101:Are Numbers Real?
1917:{\displaystyle k}
1868:{\displaystyle n}
1329:Amicable numbers
1209:An amicable pair
1145:is said to be of
1117:) is regular and
443:Thābit ibn Qurrah
409:
408:
238:(1260–1320). The
222:(826–901). Other
115:itself (see also
16:(Redirected from
5317:
5305:Divisor function
5277:
5240:
5209:Natural language
5204:
5168:
5136:Generated via a
5132:
5092:
4997:Digit-reassembly
4962:Self-descriptive
4766:
4731:
4717:
4668:Lucas–Carmichael
4658:Harmonic divisor
4606:
4532:Sparsely totient
4507:Highly cototient
4416:Multiply perfect
4406:Highly composite
4349:
4332:
4247:
4202:
4183:Telephone number
4079:
4037:
4018:Square pyramidal
4000:Stella octangula
3925:
3791:
3782:
3774:Figurate numbers
3769:
3716:
3668:
3602:
3554:
3492:
3396:
3380:
3373:
3366:
3357:
3330:Harmonic divisor
3216:Aliquot sequence
3196:Highly composite
3120:Multiply perfect
3016:
2994:Divisor function
2960:
2953:
2946:
2937:
2931:
2929:
2928:
2916:
2907:
2905:
2904:
2883:
2881:
2880:
2871:. Archived from
2855:
2852:Report MAS-R0307
2849:
2832:
2831:
2815:
2814:
2798:
2797:
2779:
2764:
2737:
2728:Amicable Numbers
2716:
2714:
2713:
2697:
2696:
2681:
2675:
2672:
2666:
2665:
2647:
2638:(107): 545–548.
2623:
2617:
2616:
2614:
2613:
2557:
2551:
2550:
2548:
2526:(1–2): 108–111.
2517:
2508:
2502:
2501:
2495:
2487:
2459:
2453:
2452:
2446:
2438:
2410:
2404:
2403:
2401:
2400:
2385:
2379:
2362:
2356:
2355:
2337:
2328:
2327:
2325:
2324:
2313:
2307:
2306:
2304:
2303:
2268:
2262:
2261:
2259:
2257:
2251:
2244:
2235:
2229:
2228:
2211:How Euler Did It
2206:
2197:
2196:
2194:
2192:
2186:
2172:(241): 489–497.
2163:
2154:
2068:Persona 4 Golden
1996:
1994:
1993:
1990:{\displaystyle }
1988:
1963:sociable numbers
1960:
1958:
1957:
1952:
1950:
1949:
1923:
1921:
1920:
1915:
1903:
1901:
1900:
1895:
1874:
1872:
1871:
1866:
1854:
1852:
1851:
1846:
1844:
1843:
1816:aliquot sequence
1805:
1803:
1802:
1797:
1746:Sociable numbers
1737:
1720:
1710:
1697:
1695:
1694:
1689:
1687:
1686:
1668:
1667:
1655:
1654:
1639:
1638:
1611:
1610:
1589:
1588:
1563:
1561:
1560:
1555:
1550:
1549:
1531:
1530:
1518:
1517:
1498:
1496:
1495:
1490:
1442:
1440:
1439:
1434:
1401:
1399:
1398:
1393:
1360:
1358:
1357:
1352:
1300:
1288:) was obtained.
1283:
1234:
1228:
1224:
1220:
1200:
1196:
1192:
1185:
1178:
1174:
1159:
1144:
1132:
1128:
1124:
1120:
1116:
1112:
1096:
1087:) is said to be
1086:
1082:
1074:
1066:
1062:
1058:
1054:
1046:
1042:
1032:
1022:
1012:
1008:
996:
984:
973:
966:
951:
943:
932:
930:
929:
924:
922:
909:
908:
890:
889:
874:
873:
832:
831:
810:
809:
768:
767:
746:
745:
684:
676:
672:
665:
661:
654:
650:
643:
639:
632:
617:
609:
600:
598:
597:
592:
590:
577:
576:
529:
528:
490:
489:
280:
261:(1601–1665) and
234:(980–1037), and
220:Thābit ibn Qurra
197:
184:sociable numbers
168:aliquot sequence
162:
152:
138:
117:divisor function
53:Amicable numbers
39:friendly numbers
21:
5325:
5324:
5320:
5319:
5318:
5316:
5315:
5314:
5290:
5289:
5288:
5283:
5261:
5257:Strobogrammatic
5248:
5230:
5212:
5194:
5176:
5158:
5140:
5122:
5099:
5078:
5062:
5021:Divisor-related
5016:
4976:
4927:
4897:
4834:
4818:
4797:
4764:
4737:
4725:
4707:
4619:
4618:related numbers
4592:
4569:
4536:
4527:Perfect totient
4493:
4470:
4401:Highly abundant
4343:
4322:
4254:
4237:
4209:
4192:
4178:Stirling second
4084:
4061:
4022:
4004:
3961:
3910:
3847:
3808:Centered square
3776:
3759:
3721:
3706:
3673:
3658:
3610:
3609:defined numbers
3592:
3559:
3544:
3515:Double Mersenne
3501:
3482:
3404:
3390:
3388:natural numbers
3384:
3354:
3349:
3293:
3249:
3210:
3181:Highly abundant
3159:
3140:Unitary perfect
3093:
3017:
3008:
2989:Unitary divisor
2967:
2964:
2926:
2924:
2919:
2910:
2902:
2900:
2886:
2878:
2876:
2858:
2847:
2842:
2839:
2819:
2818:
2802:
2801:
2788:"Amicable Pair"
2783:
2782:
2767:
2753:
2740:
2726:, ed. (1911). "
2722:
2711:
2709:
2706:
2701:
2700:
2683:
2682:
2678:
2673:
2669:
2645:10.2307/2004382
2625:
2624:
2620:
2611:
2609:
2559:
2558:
2554:
2546:
2515:
2510:
2509:
2505:
2488:
2476:10.2307/2004629
2461:
2460:
2456:
2439:
2427:10.2307/2004381
2412:
2411:
2407:
2398:
2396:
2387:
2386:
2382:
2377:Wayback Machine
2363:
2359:
2352:
2339:
2338:
2331:
2322:
2320:
2315:
2314:
2310:
2301:
2299:
2292:
2284:. p. 168.
2270:
2269:
2265:
2255:
2253:
2249:
2242:
2237:
2236:
2232:
2225:
2208:
2207:
2200:
2190:
2188:
2184:
2161:
2156:
2155:
2151:
2146:
2137:Amicable triple
2127:
2011:
1999:perfect numbers
1967:
1966:
1935:
1930:
1929:
1906:
1905:
1877:
1876:
1857:
1856:
1829:
1824:
1823:
1812:
1758:
1757:
1754:
1752:Sociable number
1748:
1733:
1716:
1706:
1703:amicable triple
1678:
1659:
1646:
1630:
1602:
1580:
1569:
1568:
1541:
1522:
1509:
1501:
1500:
1445:
1444:
1404:
1403:
1363:
1362:
1331:
1330:
1327:
1325:Amicable tuples
1322:
1320:Generalizations
1296:
1279:
1245:
1230:
1226:
1222:
1210:
1207:
1198:
1194:
1187:
1180:
1176:
1164:
1149:
1134:
1130:
1126:
1122:
1118:
1114:
1110:
1092:
1084:
1080:
1079:then the pair (
1072:
1064:
1060:
1056:
1052:
1044:
1034:
1024:
1014:
1010:
1006:
1003:
986:
975:
968:
957:
949:
934:
920:
919:
894:
881:
859:
849:
843:
842:
823:
795:
785:
779:
778:
759:
731:
721:
709:
708:
702:
694:natural integer
682:
674:
667:
663:
656:
652:
645:
641:
634:
623:
615:
604:
588:
587:
559:
546:
540:
539:
520:
507:
501:
500:
475:
462:
450:
449:
429:
417:
208:
207:
202:
199:
195:
192:
154:
144:
134:
65:proper divisors
57:natural numbers
42:
35:
28:
23:
22:
15:
12:
11:
5:
5323:
5321:
5313:
5312:
5307:
5302:
5292:
5291:
5285:
5284:
5282:
5281:
5270:
5267:
5266:
5263:
5262:
5260:
5259:
5253:
5250:
5249:
5243:
5236:
5235:
5232:
5231:
5229:
5228:
5223:
5217:
5214:
5213:
5207:
5200:
5199:
5196:
5195:
5193:
5192:
5190:Sorting number
5187:
5185:Pancake number
5181:
5178:
5177:
5171:
5164:
5163:
5160:
5159:
5157:
5156:
5151:
5145:
5142:
5141:
5135:
5128:
5127:
5124:
5123:
5121:
5120:
5115:
5110:
5104:
5101:
5100:
5097:Binary numbers
5095:
5088:
5087:
5084:
5083:
5080:
5079:
5077:
5076:
5070:
5068:
5064:
5063:
5061:
5060:
5055:
5050:
5045:
5040:
5035:
5030:
5024:
5022:
5018:
5017:
5015:
5014:
5009:
5004:
4999:
4994:
4988:
4986:
4978:
4977:
4975:
4974:
4969:
4964:
4959:
4954:
4949:
4944:
4938:
4936:
4929:
4928:
4926:
4925:
4924:
4923:
4912:
4910:
4907:P-adic numbers
4903:
4902:
4899:
4898:
4896:
4895:
4894:
4893:
4883:
4878:
4873:
4868:
4863:
4858:
4853:
4848:
4842:
4840:
4836:
4835:
4833:
4832:
4826:
4824:
4823:Coding-related
4820:
4819:
4817:
4816:
4811:
4805:
4803:
4799:
4798:
4796:
4795:
4790:
4785:
4780:
4774:
4772:
4763:
4762:
4761:
4760:
4758:Multiplicative
4755:
4744:
4742:
4727:
4726:
4722:Numeral system
4720:
4713:
4712:
4709:
4708:
4706:
4705:
4700:
4695:
4690:
4685:
4680:
4675:
4670:
4665:
4660:
4655:
4650:
4645:
4640:
4635:
4630:
4624:
4621:
4620:
4609:
4602:
4601:
4598:
4597:
4594:
4593:
4591:
4590:
4585:
4579:
4577:
4571:
4570:
4568:
4567:
4562:
4557:
4552:
4546:
4544:
4538:
4537:
4535:
4534:
4529:
4524:
4519:
4514:
4512:Highly totient
4509:
4503:
4501:
4495:
4494:
4492:
4491:
4486:
4480:
4478:
4472:
4471:
4469:
4468:
4463:
4458:
4453:
4448:
4443:
4438:
4433:
4428:
4423:
4418:
4413:
4408:
4403:
4398:
4393:
4388:
4383:
4378:
4373:
4368:
4366:Almost perfect
4363:
4357:
4355:
4345:
4344:
4335:
4328:
4327:
4324:
4323:
4321:
4320:
4315:
4310:
4305:
4300:
4295:
4290:
4285:
4280:
4275:
4270:
4265:
4259:
4256:
4255:
4250:
4243:
4242:
4239:
4238:
4236:
4235:
4230:
4225:
4220:
4214:
4211:
4210:
4205:
4198:
4197:
4194:
4193:
4191:
4190:
4185:
4180:
4175:
4173:Stirling first
4170:
4165:
4160:
4155:
4150:
4145:
4140:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4100:
4095:
4089:
4086:
4085:
4082:
4075:
4074:
4071:
4070:
4067:
4066:
4063:
4062:
4060:
4059:
4054:
4049:
4043:
4041:
4034:
4028:
4027:
4024:
4023:
4021:
4020:
4014:
4012:
4006:
4005:
4003:
4002:
3997:
3992:
3987:
3982:
3977:
3971:
3969:
3963:
3962:
3960:
3959:
3954:
3949:
3944:
3939:
3933:
3931:
3922:
3916:
3915:
3912:
3911:
3909:
3908:
3903:
3898:
3893:
3888:
3883:
3878:
3873:
3868:
3863:
3857:
3855:
3849:
3848:
3846:
3845:
3840:
3835:
3830:
3825:
3820:
3815:
3810:
3805:
3799:
3797:
3788:
3778:
3777:
3772:
3765:
3764:
3761:
3760:
3758:
3757:
3752:
3747:
3742:
3737:
3732:
3726:
3723:
3722:
3719:
3712:
3711:
3708:
3707:
3705:
3704:
3699:
3694:
3689:
3684:
3678:
3675:
3674:
3671:
3664:
3663:
3660:
3659:
3657:
3656:
3651:
3646:
3641:
3636:
3631:
3626:
3621:
3615:
3612:
3611:
3605:
3598:
3597:
3594:
3593:
3591:
3590:
3585:
3580:
3575:
3570:
3564:
3561:
3560:
3557:
3550:
3549:
3546:
3545:
3543:
3542:
3537:
3532:
3527:
3522:
3517:
3512:
3506:
3503:
3502:
3495:
3488:
3487:
3484:
3483:
3481:
3480:
3475:
3470:
3465:
3460:
3455:
3450:
3445:
3440:
3435:
3430:
3425:
3420:
3415:
3409:
3406:
3405:
3399:
3392:
3391:
3385:
3383:
3382:
3375:
3368:
3360:
3351:
3350:
3348:
3347:
3342:
3337:
3332:
3327:
3322:
3317:
3312:
3307:
3301:
3299:
3295:
3294:
3292:
3291:
3286:
3281:
3276:
3271:
3266:
3260:
3258:
3251:
3250:
3248:
3247:
3242:
3237:
3227:
3221:
3219:
3212:
3211:
3209:
3208:
3203:
3198:
3193:
3188:
3183:
3178:
3173:
3167:
3165:
3161:
3160:
3158:
3157:
3152:
3147:
3142:
3137:
3132:
3127:
3122:
3117:
3112:
3110:Almost perfect
3107:
3101:
3099:
3095:
3094:
3092:
3091:
3086:
3081:
3076:
3071:
3066:
3061:
3056:
3051:
3046:
3041:
3036:
3031:
3025:
3023:
3019:
3018:
3011:
3009:
3007:
3006:
3001:
2996:
2991:
2986:
2981:
2975:
2973:
2969:
2968:
2965:
2963:
2962:
2955:
2948:
2940:
2934:
2933:
2917:
2908:
2887:Grime, James.
2884:
2859:Grime, James.
2856:
2838:
2837:External links
2835:
2834:
2833:
2822:"Euler's Rule"
2816:
2799:
2780:
2765:
2751:
2738:
2724:Chisholm, Hugh
2705:
2702:
2699:
2698:
2676:
2667:
2618:
2572:(3): 121–127.
2552:
2503:
2454:
2405:
2380:
2366:William Dunham
2357:
2350:
2329:
2308:
2290:
2272:Martin Gardner
2263:
2230:
2223:
2198:
2148:
2147:
2145:
2142:
2141:
2140:
2134:
2126:
2123:
2122:
2121:
2108:
2097:
2088:
2081:
2072:
2063:
2050:
2039:
2029:
2010:
2007:
2003:amicable pairs
1986:
1983:
1980:
1977:
1974:
1948:
1945:
1942:
1938:
1913:
1893:
1890:
1887:
1884:
1864:
1842:
1839:
1836:
1832:
1820:directed graph
1811:
1808:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1750:Main article:
1747:
1744:
1699:
1698:
1685:
1681:
1677:
1674:
1671:
1666:
1662:
1658:
1653:
1649:
1645:
1642:
1637:
1633:
1629:
1626:
1623:
1620:
1617:
1614:
1609:
1605:
1601:
1598:
1595:
1592:
1587:
1583:
1579:
1576:
1553:
1548:
1544:
1540:
1537:
1534:
1529:
1525:
1521:
1516:
1512:
1508:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1452:
1432:
1429:
1426:
1423:
1420:
1417:
1414:
1411:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1350:
1347:
1344:
1341:
1338:
1326:
1323:
1321:
1318:
1276:Martin Gardner
1244:
1241:
1206:
1203:
1173:) = (220, 284)
1002:
999:
918:
915:
912:
907:
904:
901:
897:
893:
888:
884:
880:
877:
872:
869:
866:
862:
858:
855:
852:
850:
848:
845:
844:
841:
838:
835:
830:
826:
822:
819:
816:
813:
808:
805:
802:
798:
794:
791:
788:
786:
784:
781:
780:
777:
774:
771:
766:
762:
758:
755:
752:
749:
744:
741:
738:
734:
730:
727:
724:
722:
720:
717:
716:
701:
698:
679:Thabit numbers
653:(17296, 18416)
586:
583:
580:
575:
572:
569:
566:
562:
558:
555:
552:
549:
547:
545:
542:
541:
538:
535:
532:
527:
523:
519:
516:
513:
510:
508:
506:
503:
502:
499:
496:
493:
488:
485:
482:
478:
474:
471:
468:
465:
463:
461:
458:
457:
428:
425:
416:
413:
407:
406:
403:
400:
396:
395:
392:
389:
385:
384:
381:
378:
374:
373:
370:
367:
363:
362:
359:
356:
352:
351:
348:
345:
341:
340:
337:
334:
330:
329:
326:
323:
319:
318:
315:
312:
308:
307:
304:
301:
297:
296:
291:
286:
242:mathematician
218:mathematician
203:
200:
194:
191:
188:
176:perfect number
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5322:
5311:
5308:
5306:
5303:
5301:
5298:
5297:
5295:
5280:
5276:
5272:
5271:
5268:
5258:
5255:
5254:
5251:
5246:
5241:
5237:
5227:
5224:
5222:
5219:
5218:
5215:
5210:
5205:
5201:
5191:
5188:
5186:
5183:
5182:
5179:
5174:
5169:
5165:
5155:
5152:
5150:
5147:
5146:
5143:
5139:
5133:
5129:
5119:
5116:
5114:
5111:
5109:
5106:
5105:
5102:
5098:
5093:
5089:
5075:
5072:
5071:
5069:
5065:
5059:
5056:
5054:
5051:
5049:
5048:Polydivisible
5046:
5044:
5041:
5039:
5036:
5034:
5031:
5029:
5026:
5025:
5023:
5019:
5013:
5010:
5008:
5005:
5003:
5000:
4998:
4995:
4993:
4990:
4989:
4987:
4984:
4979:
4973:
4970:
4968:
4965:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4945:
4943:
4940:
4939:
4937:
4934:
4930:
4922:
4919:
4918:
4917:
4914:
4913:
4911:
4908:
4904:
4892:
4889:
4888:
4887:
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4867:
4864:
4862:
4859:
4857:
4854:
4852:
4849:
4847:
4844:
4843:
4841:
4837:
4831:
4828:
4827:
4825:
4821:
4815:
4812:
4810:
4807:
4806:
4804:
4802:Digit product
4800:
4794:
4791:
4789:
4786:
4784:
4781:
4779:
4776:
4775:
4773:
4771:
4767:
4759:
4756:
4754:
4751:
4750:
4749:
4746:
4745:
4743:
4741:
4736:
4732:
4728:
4723:
4718:
4714:
4704:
4701:
4699:
4696:
4694:
4691:
4689:
4686:
4684:
4681:
4679:
4676:
4674:
4671:
4669:
4666:
4664:
4661:
4659:
4656:
4654:
4651:
4649:
4646:
4644:
4641:
4639:
4638:Erdős–Nicolas
4636:
4634:
4631:
4629:
4626:
4625:
4622:
4617:
4613:
4607:
4603:
4589:
4586:
4584:
4581:
4580:
4578:
4576:
4572:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4547:
4545:
4543:
4539:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4504:
4502:
4500:
4496:
4490:
4487:
4485:
4482:
4481:
4479:
4477:
4473:
4467:
4464:
4462:
4459:
4457:
4456:Superabundant
4454:
4452:
4449:
4447:
4444:
4442:
4439:
4437:
4434:
4432:
4429:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4407:
4404:
4402:
4399:
4397:
4394:
4392:
4389:
4387:
4384:
4382:
4379:
4377:
4374:
4372:
4369:
4367:
4364:
4362:
4359:
4358:
4356:
4354:
4350:
4346:
4342:
4338:
4333:
4329:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4260:
4257:
4253:
4248:
4244:
4234:
4231:
4229:
4226:
4224:
4221:
4219:
4216:
4215:
4212:
4208:
4203:
4199:
4189:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4090:
4087:
4080:
4076:
4058:
4055:
4053:
4050:
4048:
4045:
4044:
4042:
4038:
4035:
4033:
4032:4-dimensional
4029:
4019:
4016:
4015:
4013:
4011:
4007:
4001:
3998:
3996:
3993:
3991:
3988:
3986:
3983:
3981:
3978:
3976:
3973:
3972:
3970:
3968:
3964:
3958:
3955:
3953:
3950:
3948:
3945:
3943:
3942:Centered cube
3940:
3938:
3935:
3934:
3932:
3930:
3926:
3923:
3921:
3920:3-dimensional
3917:
3907:
3904:
3902:
3899:
3897:
3894:
3892:
3889:
3887:
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3864:
3862:
3859:
3858:
3856:
3854:
3850:
3844:
3841:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3801:
3800:
3798:
3796:
3792:
3789:
3787:
3786:2-dimensional
3783:
3779:
3775:
3770:
3766:
3756:
3753:
3751:
3748:
3746:
3743:
3741:
3738:
3736:
3733:
3731:
3730:Nonhypotenuse
3728:
3727:
3724:
3717:
3713:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3679:
3676:
3669:
3665:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3630:
3627:
3625:
3622:
3620:
3617:
3616:
3613:
3608:
3603:
3599:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3565:
3562:
3555:
3551:
3541:
3538:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3507:
3504:
3499:
3493:
3489:
3479:
3476:
3474:
3471:
3469:
3468:Perfect power
3466:
3464:
3461:
3459:
3458:Seventh power
3456:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3419:
3416:
3414:
3411:
3410:
3407:
3402:
3397:
3393:
3389:
3381:
3376:
3374:
3369:
3367:
3362:
3361:
3358:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3302:
3300:
3296:
3290:
3287:
3285:
3284:Polydivisible
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3261:
3259:
3256:
3252:
3246:
3243:
3241:
3238:
3235:
3231:
3228:
3226:
3223:
3222:
3220:
3217:
3213:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3187:
3186:Superabundant
3184:
3182:
3179:
3177:
3174:
3172:
3169:
3168:
3166:
3162:
3156:
3155:Erdős–Nicolas
3153:
3151:
3148:
3146:
3143:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3106:
3103:
3102:
3100:
3096:
3090:
3087:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3065:
3064:Perfect power
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3026:
3024:
3020:
3015:
3005:
3002:
3000:
2997:
2995:
2992:
2990:
2987:
2985:
2982:
2980:
2977:
2976:
2974:
2970:
2961:
2956:
2954:
2949:
2947:
2942:
2941:
2938:
2923:
2918:
2914:
2909:
2898:
2894:
2890:
2885:
2875:on 2017-07-16
2874:
2870:
2866:
2862:
2857:
2853:
2846:
2841:
2840:
2836:
2829:
2828:
2823:
2817:
2812:
2811:
2806:
2800:
2795:
2794:
2789:
2786:
2781:
2777:
2776:Penguin Group
2773:
2772:
2766:
2762:
2758:
2754:
2748:
2744:
2739:
2735:
2734:
2729:
2725:
2720:
2719:public domain
2708:
2707:
2703:
2695:
2691:
2687:
2680:
2677:
2671:
2668:
2663:
2659:
2655:
2651:
2646:
2641:
2637:
2633:
2629:
2622:
2619:
2607:
2603:
2599:
2595:
2591:
2587:
2583:
2579:
2575:
2571:
2567:
2563:
2556:
2553:
2545:
2541:
2537:
2533:
2529:
2525:
2521:
2514:
2507:
2504:
2499:
2493:
2485:
2481:
2477:
2473:
2469:
2465:
2458:
2455:
2450:
2444:
2436:
2432:
2428:
2424:
2420:
2416:
2409:
2406:
2394:
2390:
2384:
2381:
2378:
2374:
2371:
2367:
2361:
2358:
2353:
2347:
2343:
2336:
2334:
2330:
2319:
2312:
2309:
2297:
2293:
2291:9781470463588
2287:
2283:
2279:
2278:
2273:
2267:
2264:
2248:
2241:
2234:
2231:
2226:
2220:
2216:
2212:
2205:
2203:
2199:
2183:
2179:
2175:
2171:
2167:
2160:
2153:
2150:
2143:
2138:
2135:
2132:
2129:
2128:
2124:
2119:
2115:
2114:
2109:
2106:
2102:
2098:
2095:
2094:
2089:
2086:
2082:
2079:
2078:
2073:
2070:
2069:
2064:
2061:
2057:
2056:
2051:
2048:
2047:Reginald Hill
2044:
2040:
2037:
2033:
2030:
2027:
2026:Japanese film
2024:, and in the
2023:
2019:
2018:
2013:
2012:
2008:
2006:
2004:
2000:
1981:
1978:
1975:
1964:
1946:
1943:
1940:
1936:
1927:
1911:
1888:
1882:
1862:
1840:
1837:
1834:
1830:
1821:
1817:
1809:
1807:
1793:
1787:
1781:
1775:
1769:
1763:
1753:
1745:
1743:
1741:
1736:
1731:
1726:
1724:
1719:
1714:
1709:
1704:
1683:
1679:
1675:
1672:
1669:
1664:
1660:
1656:
1651:
1647:
1643:
1635:
1631:
1624:
1621:
1618:
1615:
1607:
1603:
1596:
1593:
1585:
1581:
1574:
1567:
1566:
1565:
1546:
1542:
1538:
1535:
1532:
1527:
1523:
1519:
1514:
1510:
1486:
1483:
1480:
1477:
1471:
1465:
1462:
1456:
1450:
1430:
1427:
1424:
1421:
1415:
1409:
1389:
1386:
1383:
1380:
1374:
1368:
1345:
1342:
1339:
1324:
1319:
1317:
1314:
1312:
1308:
1304:
1299:
1294:
1289:
1287:
1282:
1277:
1272:
1270:
1265:
1263:
1259:
1255:
1250:
1243:Other results
1242:
1240:
1238:
1233:
1218:
1214:
1204:
1202:
1193:. Therefore,
1190:
1183:
1172:
1168:
1161:
1157:
1153:
1148:
1142:
1138:
1108:
1104:
1100:
1095:
1090:
1078:
1070:
1050:
1041:
1037:
1031:
1027:
1021:
1017:
1001:Regular pairs
1000:
998:
994:
990:
982:
978:
972:
965:
961:
955:
954:prime numbers
947:
941:
937:
916:
913:
910:
905:
902:
899:
895:
891:
886:
878:
875:
870:
867:
864:
860:
853:
851:
846:
839:
836:
833:
828:
824:
820:
814:
811:
806:
803:
800:
796:
789:
787:
782:
775:
772:
769:
764:
760:
756:
750:
747:
742:
739:
736:
732:
725:
723:
718:
706:
697:
695:
691:
686:
680:
677:are known as
670:
659:
648:
638:
631:
627:
621:
620:prime numbers
613:
607:
601:
584:
581:
578:
573:
570:
567:
564:
560:
556:
553:
550:
548:
543:
536:
533:
530:
525:
521:
517:
514:
511:
509:
504:
497:
494:
491:
486:
483:
480:
476:
472:
469:
466:
464:
459:
446:
444:
441:
440:mathematician
438:
434:
426:
424:
420:
414:
412:
404:
401:
398:
397:
393:
390:
387:
386:
382:
379:
376:
375:
371:
368:
365:
364:
360:
357:
354:
353:
349:
346:
343:
342:
338:
335:
332:
331:
327:
324:
321:
320:
316:
313:
310:
309:
305:
302:
299:
298:
295:
292:
290:
287:
285:
282:
281:
275:
272:
268:
264:
260:
255:
253:
249:
245:
241:
237:
233:
230:(died 1007),
229:
225:
221:
217:
213:
206:
189:
187:
185:
181:
177:
173:
169:
164:
161:
157:
151:
147:
143:). (Also see
142:
137:
131:
129:
125:
120:
118:
114:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
66:
62:
58:
54:
46:
40:
33:
19:
5012:Transposable
4876:Narcissistic
4783:Digital root
4703:Super-Poulet
4663:Jordan–Pólya
4612:prime factor
4549:
4517:Noncototient
4484:Almost prime
4466:Superperfect
4441:Refactorable
4436:Quasiperfect
4411:Hyperperfect
4252:Pseudoprimes
4223:Wall–Sun–Sun
4158:Ordered Bell
4128:Fuss–Catalan
4040:non-centered
3990:Dodecahedral
3967:non-centered
3853:non-centered
3755:Wolstenholme
3500:× 2 ± 1
3497:
3496:Of the form
3463:Eighth power
3443:Fourth power
3345:Superperfect
3340:Refactorable
3229:
3135:Superperfect
3130:Hyperperfect
3115:Quasiperfect
2999:Prime factor
2925:. Retrieved
2901:. Retrieved
2892:
2877:. Retrieved
2873:the original
2864:
2851:
2825:
2808:
2791:
2770:
2742:
2731:
2685:
2679:
2670:
2635:
2631:
2621:
2610:. Retrieved
2569:
2565:
2555:
2523:
2519:
2506:
2492:cite journal
2467:
2463:
2457:
2443:cite journal
2418:
2414:
2408:
2397:. Retrieved
2383:
2368:in a video:
2360:
2341:
2321:. Retrieved
2311:
2300:. Retrieved
2276:
2266:
2254:. Retrieved
2247:the original
2233:
2210:
2189:. Retrieved
2169:
2165:
2152:
2118:Colum McCann
2111:
2100:
2091:
2075:
2066:
2053:
2042:
2035:
2028:based on it.
2015:
1813:
1755:
1727:
1700:
1328:
1315:
1290:
1273:
1266:
1246:
1216:
1212:
1208:
1188:
1181:
1170:
1166:
1162:
1155:
1151:
1146:
1140:
1136:
1106:
1102:
1088:
1039:
1035:
1029:
1025:
1023:, and write
1019:
1015:
1004:
992:
988:
980:
976:
970:
963:
959:
939:
935:
705:Euler's rule
704:
703:
700:Euler's rule
687:
668:
657:
646:
636:
629:
625:
605:
602:
447:
432:
430:
421:
418:
410:
293:
288:
283:
256:
212:Pythagoreans
209:
179:
165:
132:
121:
112:
108:
104:
100:
96:
92:
88:
84:
80:
76:
72:
68:
52:
51:
5033:Extravagant
5028:Equidigital
4983:permutation
4942:Palindromic
4916:Automorphic
4814:Sum-product
4793:Sum-product
4748:Persistence
4643:Erdős–Woods
4565:Untouchable
4446:Semiperfect
4396:Hemiperfect
4057:Tesseractic
3995:Icosahedral
3975:Tetrahedral
3906:Dodecagonal
3607:Recursively
3478:Prime power
3453:Sixth power
3448:Fifth power
3428:Power of 10
3386:Classes of
3269:Extravagant
3264:Equidigital
3225:Untouchable
3145:Semiperfect
3125:Hemiperfect
3054:Square-free
2869:Brady Haran
2865:Numberphile
2470:: 963–968.
2421:: 539–543.
2105:Brian Clegg
2060:Denis Guedj
2032:Paul Auster
1077:square free
232:al-Baghdadi
5294:Categories
5245:Graphemics
5118:Pernicious
4972:Undulating
4947:Pandigital
4921:Trimorphic
4522:Nontotient
4371:Arithmetic
3985:Octahedral
3886:Heptagonal
3876:Pentagonal
3861:Triangular
3702:Sierpiński
3624:Jacobsthal
3423:Power of 3
3418:Power of 2
3305:Arithmetic
3298:Other sets
3257:-dependent
2927:2023-09-10
2903:2020-06-09
2879:2013-04-02
2774:. London:
2761:1079.11001
2704:References
2612:2020-09-07
2399:2016-01-31
2323:2024-05-28
2302:2023-03-18
2022:Yōko Ogawa
1961:represent
1705:(sequence
1269:Paul Erdős
1195:(220, 284)
1091:(sequence
642:(220, 284)
228:al-Majriti
5002:Parasitic
4851:Factorion
4778:Digit sum
4770:Digit sum
4588:Fortunate
4575:Primorial
4489:Semiprime
4426:Practical
4391:Descartes
4386:Deficient
4376:Betrothed
4218:Wieferich
4047:Pentatope
4010:pyramidal
3901:Decagonal
3896:Nonagonal
3891:Octagonal
3881:Hexagonal
3740:Practical
3687:Congruent
3619:Fibonacci
3583:Loeschian
3335:Descartes
3310:Deficient
3245:Betrothed
3150:Practical
3039:Semiprime
3034:Composite
2827:MathWorld
2810:MathWorld
2793:MathWorld
2654:0025-5718
2594:0036-8733
2540:253787916
2274:(2020) .
2256:21 August
2113:Apeirogon
1794:…
1791:↦
1785:↦
1779:↦
1773:↦
1767:↦
1730:multisets
1728:Amicable
1673:⋯
1625:σ
1619:⋯
1597:σ
1575:σ
1536:…
1466:σ
1451:σ
1422:−
1410:σ
1381:−
1369:σ
1274:In 1968,
1267:In 1955,
1103:irregular
1067:are both
911:−
892:×
868:−
834:−
821:×
804:−
770:−
757:×
740:−
675:3 × 2 − 1
579:−
571:−
557:×
531:−
518:×
492:−
484:−
473:×
263:Descartes
248:Descartes
236:al-Fārisī
5074:Friedman
5007:Primeval
4952:Repdigit
4909:-related
4856:Kaprekar
4830:Meertens
4753:Additive
4740:dynamics
4648:Friendly
4560:Sociable
4550:Amicable
4361:Abundant
4341:dynamics
4163:Schröder
4153:Narayana
4123:Eulerian
4113:Delannoy
4108:Dedekind
3929:centered
3795:centered
3682:Amenable
3639:Narayana
3629:Leonardo
3525:Mersenne
3473:Powerful
3413:Achilles
3320:Solitary
3315:Friendly
3240:Sociable
3230:Amicable
3218:-related
3171:Abundant
3069:Achilles
3059:Powerful
2972:Overview
2897:Archived
2606:Archived
2602:24926005
2544:Archived
2393:Archived
2373:Archived
2296:Archived
2191:19 April
2182:Archived
2125:See also
1875:, where
1361:satisfy
1293:amicable
946:integers
91:, where
18:Amicable
5247:related
5211:related
5175:related
5173:Sorting
5058:Vampire
5043:Harshad
4985:related
4957:Repunit
4871:Lychrel
4846:Dudeney
4698:Størmer
4693:Sphenic
4678:Regular
4616:divisor
4555:Perfect
4451:Sublime
4421:Perfect
4148:Motzkin
4103:Catalan
3644:Padovan
3578:Leyland
3573:Idoneal
3568:Hilbert
3540:Woodall
3325:Sublime
3279:Harshad
3105:Perfect
3089:Unusual
3079:Regular
3049:Sphenic
2984:Divisor
2893:YouTube
2721::
2662:2004382
2574:Bibcode
2484:0276167
2435:0246816
2085:Andante
2077:Rewrite
1788:1264460
1782:1305184
1776:1727636
1770:1547860
1764:1264460
1738:in the
1735:A259307
1721:in the
1718:A036471
1711:in the
1708:A125490
1307:A360054
1301:in the
1298:A291422
1284:in the
1281:A291550
1262:product
1258:coprime
1235:in the
1232:A273259
1179:and so
1097:in the
1094:A215491
1089:regular
1069:coprime
1047:is the
956:, then
950:p, q, r
622:, then
616:p, q, r
612:integer
405:66,992
394:76,084
383:18,416
372:14,595
361:10,856
240:Iranian
190:History
180:its own
160:A002046
158::
150:A002025
148::
139:in the
136:A259180
111:except
63:of the
5113:Odious
5038:Frugal
4992:Cyclic
4981:Digit-
4688:Smooth
4673:Pronic
4633:Cyclic
4610:Other
4583:Euclid
4233:Wilson
4207:Primes
3866:Square
3735:Polite
3697:Riesel
3692:Knödel
3654:Perrin
3535:Thabit
3520:Fermat
3510:Cullen
3433:Square
3401:Powers
3274:Frugal
3234:Triple
3074:Smooth
3044:Pronic
2759:
2749:
2715:
2660:
2652:
2600:
2592:
2538:
2482:
2433:
2348:
2288:
2221:
1926:Cycles
1254:factor
1199:(2, 1)
1109:. If (
1107:exotic
1043:where
942:> 0
933:where
690:lemmas
662:, and
610:is an
608:> 1
603:where
402:66,928
391:63,020
380:17,296
369:12,285
358:10,744
350:6,368
339:5,564
328:2,924
317:1,210
259:Fermat
172:period
5154:Prime
5149:Lucky
5138:sieve
5067:Other
5053:Smith
4933:Digit
4891:Happy
4866:Keith
4839:Other
4683:Rough
4653:Giuga
4118:Euler
3980:Cubic
3634:Lucas
3530:Proth
3289:Smith
3206:Weird
3084:Rough
3029:Prime
2848:(PDF)
2658:JSTOR
2598:JSTOR
2547:(PDF)
2536:S2CID
2516:(PDF)
2250:(PDF)
2243:(PDF)
2185:(PDF)
2162:(PDF)
2144:Notes
1125:have
1059:. If
1018:<
1005:Let (
938:>
347:6,232
336:5,020
325:2,620
314:1,184
271:Borho
267:Euler
216:Iraqi
5108:Evil
4788:Self
4738:and
4628:Blum
4339:and
4143:Lobb
4098:Cake
4093:Bell
3843:Star
3750:Ulam
3649:Pell
3438:Cube
3255:Base
2747:ISBN
2650:ISSN
2590:ISSN
2498:link
2449:link
2364:See
2346:ISBN
2286:ISBN
2258:2012
2219:ISBN
2193:2007
1814:The
1740:OEIS
1723:OEIS
1713:OEIS
1402:and
1311:OEIS
1303:OEIS
1286:OEIS
1249:even
1237:OEIS
1225:and
1191:= 71
1186:and
1184:= 55
1147:type
1129:and
1121:and
1099:OEIS
1075:and
1063:and
1055:and
1033:and
969:2 ×
967:and
958:2 ×
952:are
948:and
944:are
666:for
655:for
644:for
635:2 ×
633:and
624:2 ×
618:are
614:and
437:Arab
431:The
306:284
224:Arab
156:OEIS
153:and
146:OEIS
141:OEIS
99:)=σ(
79:and
5226:Ban
4614:or
4133:Lah
2757:Zbl
2730:".
2690:doi
2640:doi
2582:doi
2570:218
2528:doi
2472:doi
2423:doi
2174:doi
2116:by
2103:by
2058:by
2045:by
2020:by
1928:in
1742:).
1725:).
1313:).
1309:in
1239:).
1105:or
1071:to
1051:of
983:− 1
671:= 7
660:= 4
649:= 2
303:220
170:of
128:284
124:220
119:).
61:sum
5296::
2895:.
2891:.
2867:.
2863:.
2850:.
2824:.
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2790:.
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2656:.
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2596:.
2588:.
2580:.
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2480:MR
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2168:.
2164:.
2005:.
1924:.
1822:,
1215:,
1201:.
1169:,
1160:.
1154:,
1139:,
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1083:,
1040:gN
1038:=
1030:gM
1028:=
1009:,
979:=
962:×
685:.
651:,
628:×
445:.
399:10
186:.
126:,
103:)-
87:)=
75:)=
3498:a
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1985:]
1982:n
1979:,
1976:1
1973:[
1947:s
1944:,
1941:n
1937:G
1912:k
1892:)
1889:k
1886:(
1883:s
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1841:s
1838:,
1835:n
1831:G
1684:k
1680:n
1676:+
1670:+
1665:2
1661:n
1657:+
1652:1
1648:n
1644:=
1641:)
1636:k
1632:n
1628:(
1622:=
1616:=
1613:)
1608:2
1604:n
1600:(
1594:=
1591:)
1586:1
1582:n
1578:(
1552:)
1547:k
1543:n
1539:,
1533:,
1528:2
1524:n
1520:,
1515:1
1511:n
1507:(
1487:n
1484:+
1481:m
1478:=
1475:)
1472:n
1469:(
1463:=
1460:)
1457:m
1454:(
1431:m
1428:=
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1419:)
1416:n
1413:(
1390:n
1387:=
1384:m
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1340:m
1337:(
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1189:N
1182:M
1177:4
1171:n
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1165:(
1158:)
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1150:(
1143:)
1141:n
1137:m
1135:(
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1123:N
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1111:m
1085:n
1081:m
1073:g
1065:N
1061:M
1057:n
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1045:g
1036:n
1026:m
1020:n
1016:m
1011:n
1007:m
993:n
991:,
989:m
987:(
981:n
977:m
971:r
964:q
960:p
940:m
936:n
917:,
914:1
906:n
903:+
900:m
896:2
887:2
883:)
879:1
876:+
871:m
865:n
861:2
857:(
854:=
847:r
840:,
837:1
829:n
825:2
818:)
815:1
812:+
807:m
801:n
797:2
793:(
790:=
783:q
776:,
773:1
765:m
761:2
754:)
751:1
748:+
743:m
737:n
733:2
729:(
726:=
719:p
683:n
669:n
658:n
647:n
637:r
630:q
626:p
606:n
585:,
582:1
574:1
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565:2
561:2
554:9
551:=
544:r
537:,
534:1
526:n
522:2
515:3
512:=
505:q
498:,
495:1
487:1
481:n
477:2
470:3
467:=
460:p
388:9
377:8
366:7
355:6
344:5
333:4
322:3
311:2
300:1
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289:m
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