44:
5333:
2384:
392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496, 536, 556, 563, 565, 566, 608, 617, 622, 623, 632, 635, 637, 638, 644, 649, 653, 655, 656, 665, 671, 673, 680, 683, 694, 700, 709, 716, 736, 739, 748, 761, 763, 784, 790, 793, 802, 806, 818, 820, 833, 836, 847, 860, 863, 874, 881, 888, 899, 901, 904, 907, 910, 912, 913, 921, 923, 931, 932, 937, 940, 946, 964, 970, 973, 989, 998, 1000 (sequence
2266:
1, 6, 36, 44, 49, 79, 100, 160, 170, 216, 224, 229, 254, 264, 275, 285, 289, 294, 335, 347, 355, 357, 388, 405, 415, 417, 439, 460, 469, 474, 533, 538, 580, 593, 600, 608, 628, 638, 647, 695, 707, 715, 717, 767, 777, 787, 835, 837, 847, 880, 890, 928, 940, 953, 960, 968, 1010, 1018, 1020, 1033, 1058,
2642:
11031, 1233E, 13011, 1332E, 16377, 17367, 17637, 22E8E, 2331E, 233E1, 23955, 25935, 25X8E, 28X5E, 28XE5, 2X8E5, 2E82E, 2E8X5, 31011, 31101, 3123E, 3132E, 31677, 33E21, 35295, 35567, 35765, 35925, 36557, 37167, 37671, 39525, 4878E, 4X7X7, 53567, 55367, 55637, 56357, 57635, 58XX5, 5X82E, 5XX85, 606EE,
2383:
1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391,
2643:
63575, 63771, 66E0E, 67317, 67371, 67535, 6E60E, 71367, 71637, 73167, 76137, 7XX47, 82XE5, 82EX5, 8487E, 848E7, 84E87, 8874E, 8X1X7, 8X25E, 8X2E5, 8X5X5, 8XX17, 8XX71, 8E2X5, 8E847, 92355, 93255, 93525, 95235, X1X87, X258E, X285E, X2E85, X85X5, X8X17, XX477, XX585, E228E, E606E, E822E, EX825, ...
594:
2416:
The first pair of consecutive 10-happy numbers is 31 and 32. The first set of three consecutive is 1880, 1881, and 1882. It has been proven that there exist sequences of consecutive happy numbers of any natural number length. The beginning of the first run of at least
2376:, all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 10 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits.
2259:, all numbers either lead to 1 and are happy, or lead to the cycle and are unhappy. Because base 6 has no other perfect digital invariants except for 1, no positive integer other than 1 is the sum of the squares of its own digits.
2445:
3, 20, 143, 1442, 14377, 143071, 1418854, 14255667, 145674808, 1492609148, 15091199357, 149121303586, 1443278000870, 13770853279685, 130660965862333, 1245219117260664, 12024696404768025, 118226055080025491, 1183229962059381238,
2433:
As Robert Styer puts it in his paper calculating this series: "Amazingly, the same value of N that begins the least sequence of six consecutive happy numbers also begins the least sequence of seven consecutive happy numbers."
1623:
1766:
1467:
430:
1326:
1945:
of around 0.15. Perhaps surprisingly, then, the 10-happy numbers do not have an asymptotic density. The upper density of the happy numbers is greater than 0.18577, and the lower density is less than 0.1138.
1205:
1097:
989:
3435:
2519:-happy prime will not necessarily create another happy prime. For instance, while 19 is a 10-happy prime, 91 = 13 Ă 7 is not prime (but is still 10-happy).
737:
228:
103:
274:
149:
775:
6137:
182:
2374:
2333:
2257:
2216:
2145:
2108:
889:
832:
1883:
624:
1856:
2702:
2300:
5369:
2676:
2183:
2075:
366:
5740:
2517:
2493:
2473:
2016:
1996:
1923:
1903:
1829:
1809:
1789:
909:
852:
795:
684:
664:
644:
419:
336:
305:
2704:
described in the definition of happy given at the top of this article, repeatedly; after each time, they check for both halt conditions: reaching 1, and
2638:, there are no 12-happy primes less than 10000, the first 12-happy primes are (the letters X and E represent the decimal numbers 10 and 11 respectively)
276:, the number that started the sequence, and so the process continues in an infinite cycle without ever reaching 1. A number which is not happy is called
1473:
6643:
3246:
3187:
3129:
3096:
3071:
2570:
2409:
2391:
5822:
3428:
2402:
1, 7, 13, 19, 23, 28, 44, 49, 68, 79, 129, 133, 139, 167, 188, 226, 236, 239, 338, 356, 367, 368, 379, 446, 469, 478, 556, 566, 888, 899. (sequence
1972:
2398:
The distinct combinations of digits that form 10-happy numbers below 1000 are (the rest are just rearrangements and/or insertions of zero digits):
5745:
1629:
1337:
5659:
3336:
589:{\displaystyle F_{p,b}(n)=\sum _{i=0}^{\lfloor \log _{b}{n}\rfloor }{\left({\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}\right)}^{p}}
4235:
3421:
1211:
59:
is a number which eventually reaches 1 when replaced by the sum of the square of each digit. For instance, 13 is a happy number because
5362:
4230:
4245:
4225:
3319:
1103:
995:
5996:
4938:
4518:
6077:
4240:
5024:
3386:
5355:
3378:
2712:
6199:
5857:
4340:
6638:
4690:
4009:
3802:
3208:
6224:
4725:
4695:
4370:
4360:
5690:
4866:
4280:
4014:
3994:
3363:
4556:
4720:
2563:
7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487 (sequence
6132:
4815:
4438:
4195:
4004:
3986:
3880:
3870:
3860:
4700:
4943:
4488:
4109:
3895:
3890:
3885:
3875:
3852:
2982:
1929:
of a number is preserved by removing or inserting zeroes at will, since they do not contribute to the cross sum.
920:
422:
396:
339:
3928:
4185:
5765:
5054:
5019:
4805:
4715:
4589:
4564:
4473:
4463:
4075:
4057:
3977:
6282:
5411:
5314:
4584:
4458:
4089:
3865:
3645:
3572:
3217:
6619:
6209:
5862:
5770:
4569:
4423:
4350:
3505:
5278:
4918:
689:
383:) by his daughter, who had learned of them at school. However, they "may have originated in Russia" (
6189:
5211:
5105:
5069:
4810:
4533:
4513:
4330:
3999:
3787:
3759:
3043:
187:
62:
6184:
5842:
4933:
4797:
4792:
4760:
4523:
4498:
4493:
4468:
4398:
4394:
4325:
4215:
4047:
3843:
3812:
2962:
2110:
is the trivial perfect digital invariant 1, and there are no other cycles. Because all numbers are
233:
108:
5332:
6292:
6229:
6219:
6204:
5837:
5695:
5336:
5090:
5085:
4999:
4973:
4871:
4850:
4622:
4503:
4453:
4375:
4345:
4285:
4052:
4032:
3963:
3676:
3153:
3033:
742:
372:
5616:
4220:
3203:
6261:
6236:
6214:
6194:
5817:
5789:
5482:
5230:
5175:
5029:
5004:
4978:
4755:
4433:
4428:
4355:
4335:
4320:
4042:
4024:
3943:
3933:
3918:
3696:
3681:
3343:
3315:
2578:
798:
154:
2346:
2305:
2229:
2188:
2117:
2080:
861:
804:
6171:
6161:
6156:
6093:
5940:
5807:
5710:
5266:
5059:
4645:
4617:
4607:
4599:
4483:
4448:
4443:
4410:
4104:
4067:
3958:
3953:
3948:
3938:
3910:
3797:
3749:
3744:
3701:
3640:
2967:
1861:
603:
380:
376:
1834:
5872:
5832:
5715:
5680:
5644:
5599:
5452:
5440:
5242:
5131:
5064:
4990:
4913:
4887:
4705:
4418:
4275:
4210:
4180:
4170:
4165:
3831:
3739:
3686:
3530:
3470:
3367:
3311:
2705:
2681:
2279:
1942:
371:
The origin of happy numbers is not clear. Happy numbers were brought to the attention of
2655:
2335:
is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle
2218:
is the trivial perfect digital invariant 1, and the only cycle is the eight-number cycle
2162:
2054:
345:
3047:
2861:"""Determine if the specified number is a happy number."""
6277:
6251:
6148:
6016:
5867:
5827:
5812:
5684:
5575:
5540:
5495:
5420:
5402:
5247:
5115:
5100:
4964:
4928:
4903:
4779:
4750:
4735:
4612:
4508:
4478:
4205:
4160:
4037:
3635:
3630:
3625:
3597:
3582:
3495:
3480:
3458:
3445:
3303:
2972:
2613:
2502:
2478:
2458:
2111:
2001:
1981:
1941:
By inspection of the first million or so 10-happy numbers, it appears that they have a
1908:
1888:
1814:
1794:
1774:
894:
837:
780:
669:
649:
629:
404:
321:
312:
290:
31:
3372:
6632:
6287:
6052:
5916:
5889:
5725:
5590:
5528:
5519:
5504:
5467:
5393:
5170:
5154:
5095:
5049:
4745:
4730:
4640:
4365:
3923:
3792:
3754:
3711:
3592:
3577:
3567:
3525:
3515:
3490:
52:
43:
6608:
6603:
6598:
6593:
6588:
6583:
6578:
6573:
6568:
6563:
6558:
6553:
6548:
6543:
6538:
6533:
6528:
6523:
6518:
6513:
6508:
6503:
6498:
6493:
6488:
6483:
6478:
6473:
6468:
6463:
6458:
6453:
6448:
6443:
6438:
6241:
5964:
5730:
5720:
5705:
5700:
5664:
5378:
5206:
5195:
5110:
4923:
4840:
4740:
4710:
4685:
4669:
4574:
4541:
4290:
4264:
4175:
4114:
3691:
3587:
3520:
3500:
3475:
2977:
2609:= 176, which is a 10-happy number. Paul Jobling discovered the prime in 2005.
2496:
2429:
1, 31, 1880, 7839, 44488, 7899999999999959999999996, 7899999999999959999999996, ...
3346:
3281:
3262:
1618:{\displaystyle F_{2,6}^{2}(347)=F_{2,6}(44)=F_{2,6}(112_{6})=1^{2}+1^{2}+2^{2}=6}
6433:
6428:
6423:
6418:
6413:
6408:
6403:
6398:
6393:
6388:
6383:
6378:
6373:
6368:
6363:
6358:
6353:
6348:
6343:
6338:
6333:
6179:
5852:
5760:
5755:
5735:
5649:
5552:
5428:
5165:
5040:
4845:
4309:
4200:
4155:
4150:
3900:
3807:
3706:
3535:
3510:
3485:
3398:
3394:
3236:
3173:
3119:
3086:
3061:
316:
151:. On the other hand, 4 is not a happy number because the sequence starting with
2605:
digits because the many 0s do not contribute to the sum of squared digits, and
6256:
6072:
5980:
5900:
5750:
5654:
5302:
5283:
4579:
4190:
3002:
3413:
6297:
6246:
6127:
4908:
4835:
4827:
4632:
4546:
3664:
3351:
3124:"Sequence A072494 (First of triples of consecutive happy numbers)"
2267:
1125, 1135, 1137, 1168, 1178, 1187, 1195, 1197, 1207, 1238, 1277, 1292, 1295
2262:
In base 10, the 74 6-happy numbers up to 1296 = 6 are (written in base 10):
3360:
17:
5009:
3091:"Sequence A035502 (Lower of pair of consecutive happy numbers)"
1761:{\displaystyle F_{2,6}^{3}(347)=F_{2,6}(6)=F_{2,6}(10_{6})=1^{2}+0^{2}=1}
1462:{\displaystyle F_{2,6}(347)=F_{2,6}(1335_{6})=1^{2}+3^{2}+3^{2}+5^{2}=44}
2652:
The examples below implement the perfect digital invariant function for
5799:
5347:
5014:
4673:
2753:"""Perfect digital invariant function."""
2635:
2556:
1953:
47:
Tree showing all happy numbers up to 100, with 130 seen with 13 and 31.
3158:
2539:
2527:
2523:
2148:
2031:
2027:
1966:
1962:
2612:
As of 2010, the largest known 10-happy prime is 2 â 1 (a
5794:
5780:
3038:
1321:{\displaystyle F_{2,10}^{4}(19)=F_{2,10}(100)=1^{2}+0^{2}+0^{2}=1}
42:
3241:"Sequence A068571 (Number of happy numbers <= 10^n)"
3178:"Sequence A055629 (Beginning of first run of at least
3066:"Sequence A161872 (Smallest unhappy number in base n)"
5351:
5300:
5264:
5228:
5192:
5152:
4777:
4666:
4392:
4307:
4262:
4139:
3829:
3776:
3728:
3662:
3614:
3552:
3456:
3417:
2546:
211, 1021, 1335, 2011, 2425, 2555, 3351, 4225, 4441, 5255, 5525
1200:{\displaystyle F_{2,10}^{3}(19)=F_{2,10}(68)=6^{2}+8^{2}=100}
546:
517:
27:
Numbers with a certain property involving recursive summation
1092:{\displaystyle F_{2,10}^{2}(19)=F_{2,10}(82)=8^{2}+2^{2}=68}
3240:
3204:"Smallest Examples of Strings of Consecutive Happy Numbers"
3177:
3123:
3090:
3065:
2565:
2404:
2386:
3024:
Gilmer, Justin (2013). "On the
Density of Happy Numbers".
2437:
The number of 10-happy numbers up to 10 for 1 â€
6328:
6323:
6318:
6313:
3223:
2684:
2658:
2522:
All prime numbers are 2-happy and 4-happy primes, as
2505:
2481:
2461:
2379:
In base 10, the 143 10-happy numbers up to 1000 are:
2349:
2308:
2282:
2232:
2191:
2165:
2120:
2083:
2057:
2004:
1984:
1911:
1891:
1864:
1837:
1817:
1797:
1777:
1632:
1476:
1340:
1214:
1106:
998:
923:
897:
864:
840:
807:
783:
745:
692:
672:
652:
632:
606:
433:
407:
348:
324:
293:
236:
190:
157:
111:
65:
2499:. Unlike happy numbers, rearranging the digits of a
2147:, all numbers lead to 1 and are happy. As a result,
6306:
6270:
6170:
6147:
6121:
5888:
5881:
5779:
5673:
5637:
5386:
5124:
5078:
5038:
4989:
4963:
4896:
4880:
4859:
4826:
4791:
4631:
4598:
4555:
4532:
4409:
4097:
4088:
4066:
4023:
3985:
3976:
3909:
3851:
3842:
2343:and because all numbers are preperiodic points for
2226:and because all numbers are preperiodic points for
2696:
2670:
2511:
2487:
2467:
2368:
2327:
2302:, the only positive perfect digital invariant for
2294:
2251:
2210:
2185:, the only positive perfect digital invariant for
2177:
2139:
2102:
2077:, the only positive perfect digital invariant for
2069:
2010:
1990:
1917:
1897:
1877:
1850:
1823:
1803:
1783:
1760:
1617:
1461:
1320:
1199:
1091:
983:
903:
883:
846:
826:
789:
769:
731:
678:
658:
638:
618:
588:
413:
360:
330:
299:
268:
222:
176:
143:
97:
338:that eventually reaches 1 when iterated over the
6010: = 0, 1, 2, 3, ...
2339:4 â 16 â 37 â 58 â 89 â 145 â 42 â 20 â 4 â ...
2222:5 â 41 â 25 â 45 â 105 â 42 â 32 â 21 â 5 â ...
2018:-happy. The only happy integer bases less than
3152:Pan, Hao (2006). "Consecutive Happy Numbers".
855:
5363:
3429:
8:
499:
478:
391:Happy numbers and perfect digital invariants
3263:"The Prime Database: 10 + 7426247 · 10 + 1"
984:{\displaystyle F_{2,10}(19)=1^{2}+9^{2}=82}
5885:
5370:
5356:
5348:
5297:
5261:
5225:
5189:
5149:
4823:
4788:
4774:
4663:
4406:
4389:
4304:
4259:
4136:
4094:
3982:
3848:
3839:
3826:
3773:
3730:Possessing a specific set of other numbers
3725:
3659:
3611:
3549:
3453:
3436:
3422:
3414:
3247:On-Line Encyclopedia of Integer Sequences
3188:On-Line Encyclopedia of Integer Sequences
3157:
3130:On-Line Encyclopedia of Integer Sequences
3097:On-Line Encyclopedia of Integer Sequences
3072:On-Line Encyclopedia of Integer Sequences
3037:
2683:
2657:
2504:
2480:
2460:
2354:
2348:
2313:
2307:
2281:
2237:
2231:
2196:
2190:
2164:
2125:
2119:
2088:
2082:
2056:
2003:
1983:
1910:
1890:
1869:
1863:
1842:
1836:
1816:
1796:
1776:
1746:
1733:
1717:
1698:
1670:
1648:
1637:
1631:
1603:
1590:
1577:
1561:
1542:
1514:
1492:
1481:
1475:
1447:
1434:
1421:
1408:
1392:
1373:
1345:
1339:
1306:
1293:
1280:
1252:
1230:
1219:
1213:
1185:
1172:
1144:
1122:
1111:
1105:
1077:
1064:
1036:
1014:
1003:
997:
969:
956:
928:
922:
896:
869:
863:
839:
812:
806:
782:
761:
750:
744:
708:
697:
691:
671:
651:
631:
605:
580:
567:
554:
549:
545:
525:
520:
516:
510:
505:
494:
485:
477:
466:
438:
432:
406:
375:(a British author and senior lecturer in
347:
323:
292:
254:
241:
235:
208:
195:
189:
162:
156:
129:
116:
110:
83:
70:
64:
2542:, the 6-happy primes below 1296 = 6 are
2994:
1973:(more unsolved problems in mathematics)
2475:-happy prime is a number that is both
854:-unhappy otherwise. If a number is a
7:
2559:, the 10-happy primes below 500 are
856:nontrivial perfect digital invariant
384:
3308:Unsolved Problems in Number Theory
423:perfect digital invariant function
340:perfect digital invariant function
25:
5847:
2607:1 + 7 + 4 + 2 + 6 + 2 + 4 + 7 + 1
2421:consecutive 10-happy numbers for
6644:Base-dependent integer sequences
5746:Supersingular (moonshine theory)
5331:
4939:Perfect digit-to-digit invariant
1925:-happy, since its sum is 1. The
1331:For example, 347 is 6-happy, as
914:For example, 19 is 10-happy, as
732:{\displaystyle F_{2,b}^{j}(n)=1}
2715:to check if a number is happy:
1956:Unsolved problem in mathematics
421:be a natural number. Given the
5741:Supersingular (elliptic curve)
3361:calculate if a number is happy
2425: = 1, 2, 3, ... is
1978:A happy base is a number base
1969:the only bases that are happy?
1723:
1710:
1688:
1682:
1660:
1654:
1567:
1554:
1532:
1526:
1504:
1498:
1398:
1385:
1363:
1357:
1270:
1264:
1242:
1236:
1162:
1156:
1134:
1128:
1054:
1048:
1026:
1020:
946:
940:
720:
714:
456:
450:
223:{\displaystyle 1^{2}+6^{2}=37}
98:{\displaystyle 1^{2}+3^{2}=10}
1:
5522:2 ± 2 ± 1
3778:Expressible via specific sums
2616:). Its decimal expansion has
1811:-happy number, and for every
269:{\displaystyle 2^{2}+0^{2}=4}
144:{\displaystyle 1^{2}+0^{2}=1}
3209:Journal of Integer Sequences
4867:Multiplicative digital root
3282:"The Prime Database: 2 â 1"
3182:consecutive happy numbers)"
770:{\displaystyle F_{2,b}^{j}}
6660:
3237:Sloane, N. J. A.
3174:Sloane, N. J. A.
3120:Sloane, N. J. A.
3087:Sloane, N. J. A.
3062:Sloane, N. J. A.
1791:-happy numbers, as 1 is a
1771:There are infinitely many
394:
29:
6617:
5327:
5310:
5296:
5274:
5260:
5238:
5224:
5202:
5188:
5161:
5148:
4944:Perfect digital invariant
4787:
4773:
4681:
4662:
4519:Superior highly composite
4405:
4388:
4316:
4303:
4271:
4258:
4146:
4135:
3838:
3825:
3783:
3772:
3735:
3724:
3672:
3658:
3621:
3610:
3563:
3548:
3466:
3452:
2983:Perfect digital invariant
2598:is a 10-happy prime with
666:-happy if there exists a
397:Perfect digital invariant
6128:Mega (1,000,000+ digits)
5997:Arithmetic progression (
4557:Euler's totient function
4341:EulerâJacobi pseudoprime
3616:Other polynomial numbers
3216:: 5. 10.6.3 – via
2717:
177:{\displaystyle 4^{2}=16}
30:Not to be confused with
4371:SomerâLucas pseudoprime
4361:LucasâCarmichael number
4196:Lazy caterer's sequence
3366:25 January 2019 at the
3337:Mathews: Happy Numbers.
3005:. Wolfram Research, Inc
2369:{\displaystyle F_{2,b}}
2328:{\displaystyle F_{2,b}}
2252:{\displaystyle F_{2,b}}
2211:{\displaystyle F_{2,b}}
2140:{\displaystyle F_{2,b}}
2103:{\displaystyle F_{2,b}}
884:{\displaystyle F_{2,b}}
827:{\displaystyle F_{2,b}}
34:(derived from Sanskrit
6283:Industrial-grade prime
5660:NewmanâShanksâWilliams
4246:WedderburnâEtherington
3646:Lucky numbers of Euler
3218:University of Waterloo
3202:Styer, Robert (2010).
2698:
2672:
2513:
2489:
2469:
2370:
2329:
2296:
2253:
2212:
2179:
2141:
2104:
2071:
2012:
1998:where every number is
1992:
1919:
1899:
1879:
1878:{\displaystyle 10^{n}}
1852:
1825:
1805:
1785:
1762:
1619:
1463:
1322:
1201:
1093:
985:
905:
885:
848:
828:
791:
771:
733:
680:
660:
640:
620:
619:{\displaystyle b>1}
590:
503:
415:
362:
332:
301:
270:
224:
178:
145:
99:
48:
6620:List of prime numbers
6078:Sophie Germain/Safe (
4534:Prime omega functions
4351:Frobenius pseudoprime
4141:Combinatorial numbers
4010:Centered dodecahedral
3803:Primary pseudoperfect
3387:"7 and Happy Numbers"
3379:145 and the Melancoil
2699:
2673:
2514:
2490:
2470:
2446:12005034444292997294.
2371:
2330:
2297:
2254:
2213:
2180:
2142:
2105:
2072:
2013:
1993:
1920:
1900:
1880:
1853:
1851:{\displaystyle b^{n}}
1826:
1806:
1786:
1763:
1620:
1464:
1323:
1202:
1094:
986:
906:
886:
849:
829:
792:
772:
734:
681:
661:
641:
621:
591:
462:
416:
363:
333:
302:
271:
225:
179:
146:
100:
46:
38:meaning "great joy").
5802:(10 â 1)/9
4993:-composition related
4793:Arithmetic functions
4395:Arithmetic functions
4331:Elliptic pseudoprime
4015:Centered icosahedral
3995:Centered tetrahedral
2697:{\displaystyle b=10}
2682:
2656:
2503:
2479:
2459:
2347:
2306:
2295:{\displaystyle b=10}
2280:
2230:
2189:
2163:
2118:
2081:
2055:
2002:
1982:
1909:
1889:
1862:
1835:
1815:
1795:
1775:
1630:
1474:
1338:
1212:
1104:
996:
921:
895:
862:
838:
805:
781:
743:
690:
670:
650:
630:
604:
431:
405:
346:
322:
291:
234:
188:
155:
109:
63:
6639:Arithmetic dynamics
6111: ± 7, ...
5638:By integer sequence
5423:(2 + 1)/3
4919:Kaprekar's constant
4439:Colossally abundant
4326:Catalan pseudoprime
4226:SchröderâHipparchus
4005:Centered octahedral
3881:Centered heptagonal
3871:Centered pentagonal
3861:Centered triangular
3461:and related numbers
3335:Schneider, Walter:
3280:Chris K. Caldwell.
3261:Chris K. Caldwell.
3048:2011arXiv1110.3836G
2963:Arithmetic dynamics
2678:and a default base
2671:{\displaystyle p=2}
2648:Programming example
2441: †20 is
2178:{\displaystyle b=6}
2070:{\displaystyle b=4}
1933:Natural density of
1653:
1497:
1235:
1127:
1019:
766:
713:
361:{\displaystyle p=2}
230:eventually reaches
6293:Formula for primes
5926: + 2 or
5858:SmarandacheâWellin
5337:Mathematics portal
5279:Aronson's sequence
5025:SmarandacheâWellin
4782:-dependent numbers
4489:Primitive abundant
4376:Strong pseudoprime
4366:Perrin pseudoprime
4346:Fermat pseudoprime
4286:Wolstenholme prime
4110:Squared triangular
3896:Centered decagonal
3891:Centered nonagonal
3886:Centered octagonal
3876:Centered hexagonal
3401:on 15 January 2018
3375:at The Math Forum.
3344:Weisstein, Eric W.
3250:. OEIS Foundation.
3191:. OEIS Foundation.
3075:. OEIS Foundation.
2706:repeating a number
2694:
2668:
2509:
2485:
2465:
2366:
2325:
2292:
2249:
2208:
2175:
2137:
2112:preperiodic points
2100:
2067:
2008:
1988:
1915:
1895:
1875:
1848:
1821:
1801:
1781:
1758:
1633:
1615:
1477:
1459:
1318:
1215:
1197:
1107:
1089:
999:
981:
901:
881:
844:
824:
787:
767:
746:
729:
693:
676:
656:
636:
616:
586:
411:
358:
328:
297:
287:More generally, a
266:
220:
174:
141:
95:
49:
6626:
6625:
6237:Carmichael number
6172:Composite numbers
6107: ± 3, 8
6103: ± 1, 4
6066: ± 1, âŠ
6062: ± 1, 4
6058: ± 1, 2
6048:
6047:
5593:3·2 â 1
5498:2·3 + 1
5412:Double Mersenne (
5345:
5344:
5323:
5322:
5292:
5291:
5256:
5255:
5220:
5219:
5184:
5183:
5144:
5143:
5140:
5139:
4959:
4958:
4769:
4768:
4658:
4657:
4654:
4653:
4600:Aliquot sequences
4411:Divisor functions
4384:
4383:
4356:Lucas pseudoprime
4336:Euler pseudoprime
4321:Carmichael number
4299:
4298:
4254:
4253:
4131:
4130:
4127:
4126:
4123:
4122:
4084:
4083:
3972:
3971:
3929:Square triangular
3821:
3820:
3768:
3767:
3720:
3719:
3654:
3653:
3606:
3605:
3544:
3543:
3133:. OEIS Foundation
3100:. OEIS Foundation
2711:A simple test in
2579:palindromic prime
2530:are happy bases.
2512:{\displaystyle b}
2488:{\displaystyle b}
2468:{\displaystyle b}
2151:is a happy base.
2011:{\displaystyle b}
1991:{\displaystyle b}
1918:{\displaystyle b}
1898:{\displaystyle b}
1824:{\displaystyle n}
1804:{\displaystyle b}
1784:{\displaystyle b}
904:{\displaystyle b}
847:{\displaystyle b}
790:{\displaystyle j}
679:{\displaystyle j}
659:{\displaystyle b}
639:{\displaystyle n}
573:
414:{\displaystyle n}
331:{\displaystyle b}
300:{\displaystyle b}
16:(Redirected from
6651:
6157:Eisenstein prime
6112:
6088:
6067:
6039:
6011:
5991:
5975:
5959:
5954: + 6,
5950: + 2,
5935:
5930: + 4,
5911:
5886:
5803:
5766:Highly cototient
5628:
5627:
5621:
5611:
5594:
5585:
5570:
5547:
5546:·2 â 1
5535:
5534:·2 + 1
5523:
5514:
5499:
5490:
5477:
5462:
5447:
5435:
5434:·2 + 1
5424:
5415:
5406:
5397:
5372:
5365:
5358:
5349:
5335:
5298:
5267:Natural language
5262:
5226:
5194:Generated via a
5190:
5150:
5055:Digit-reassembly
5020:Self-descriptive
4824:
4789:
4775:
4726:LucasâCarmichael
4716:Harmonic divisor
4664:
4590:Sparsely totient
4565:Highly cototient
4474:Multiply perfect
4464:Highly composite
4407:
4390:
4305:
4260:
4241:Telephone number
4137:
4095:
4076:Square pyramidal
4058:Stella octangula
3983:
3849:
3840:
3832:Figurate numbers
3827:
3774:
3726:
3660:
3612:
3550:
3454:
3438:
3431:
3424:
3415:
3410:
3408:
3406:
3397:. Archived from
3357:
3356:
3325:
3310:(3rd ed.).
3290:
3289:
3277:
3271:
3270:
3258:
3252:
3251:
3233:
3227:
3224:Sloane "A055629"
3221:
3199:
3193:
3192:
3170:
3164:
3163:
3161:
3149:
3143:
3142:
3140:
3138:
3116:
3110:
3109:
3107:
3105:
3083:
3077:
3076:
3058:
3052:
3051:
3041:
3021:
3015:
3014:
3012:
3010:
2999:
2968:Fortunate number
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2799:
2796:
2793:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2703:
2701:
2700:
2695:
2677:
2675:
2674:
2669:
2625:
2624:
2621:
2608:
2604:
2603:
2597:
2595:
2593:
2590:
2587:
2568:
2518:
2516:
2515:
2510:
2494:
2492:
2491:
2486:
2474:
2472:
2471:
2466:
2407:
2389:
2375:
2373:
2372:
2367:
2365:
2364:
2334:
2332:
2331:
2326:
2324:
2323:
2301:
2299:
2298:
2293:
2272:10-happy numbers
2258:
2256:
2255:
2250:
2248:
2247:
2217:
2215:
2214:
2209:
2207:
2206:
2184:
2182:
2181:
2176:
2146:
2144:
2143:
2138:
2136:
2135:
2109:
2107:
2106:
2101:
2099:
2098:
2076:
2074:
2073:
2068:
2025:
2023:
2017:
2015:
2014:
2009:
1997:
1995:
1994:
1989:
1957:
1924:
1922:
1921:
1916:
1904:
1902:
1901:
1896:
1884:
1882:
1881:
1876:
1874:
1873:
1857:
1855:
1854:
1849:
1847:
1846:
1830:
1828:
1827:
1822:
1810:
1808:
1807:
1802:
1790:
1788:
1787:
1782:
1767:
1765:
1764:
1759:
1751:
1750:
1738:
1737:
1722:
1721:
1709:
1708:
1681:
1680:
1652:
1647:
1624:
1622:
1621:
1616:
1608:
1607:
1595:
1594:
1582:
1581:
1566:
1565:
1553:
1552:
1525:
1524:
1496:
1491:
1468:
1466:
1465:
1460:
1452:
1451:
1439:
1438:
1426:
1425:
1413:
1412:
1397:
1396:
1384:
1383:
1356:
1355:
1327:
1325:
1324:
1319:
1311:
1310:
1298:
1297:
1285:
1284:
1263:
1262:
1234:
1229:
1206:
1204:
1203:
1198:
1190:
1189:
1177:
1176:
1155:
1154:
1126:
1121:
1098:
1096:
1095:
1090:
1082:
1081:
1069:
1068:
1047:
1046:
1018:
1013:
990:
988:
987:
982:
974:
973:
961:
960:
939:
938:
910:
908:
907:
902:
890:
888:
887:
882:
880:
879:
853:
851:
850:
845:
833:
831:
830:
825:
823:
822:
796:
794:
793:
788:
776:
774:
773:
768:
765:
760:
738:
736:
735:
730:
712:
707:
685:
683:
682:
677:
665:
663:
662:
657:
645:
643:
642:
637:
625:
623:
622:
617:
595:
593:
592:
587:
585:
584:
579:
578:
574:
572:
571:
562:
561:
560:
559:
558:
538:
537:
536:
535:
511:
502:
498:
490:
489:
476:
449:
448:
420:
418:
417:
412:
381:Leeds University
377:pure mathematics
367:
365:
364:
359:
337:
335:
334:
329:
306:
304:
303:
298:
275:
273:
272:
267:
259:
258:
246:
245:
229:
227:
226:
221:
213:
212:
200:
199:
183:
181:
180:
175:
167:
166:
150:
148:
147:
142:
134:
133:
121:
120:
104:
102:
101:
96:
88:
87:
75:
74:
21:
6659:
6658:
6654:
6653:
6652:
6650:
6649:
6648:
6629:
6628:
6627:
6622:
6613:
6307:First 60 primes
6302:
6266:
6166:
6149:Complex numbers
6143:
6117:
6095:
6079:
6054:
6053:Bi-twin chain (
6044:
6018:
5998:
5982:
5966:
5942:
5918:
5902:
5877:
5863:Strobogrammatic
5801:
5775:
5669:
5633:
5625:
5619:
5618:
5601:
5592:
5577:
5554:
5542:
5530:
5521:
5506:
5497:
5484:
5476:# + 1
5474:
5469:
5461:# ± 1
5459:
5454:
5446:! ± 1
5442:
5430:
5422:
5414:2 â 1
5413:
5405:2 â 1
5404:
5396:2 + 1
5395:
5382:
5376:
5346:
5341:
5319:
5315:Strobogrammatic
5306:
5288:
5270:
5252:
5234:
5216:
5198:
5180:
5157:
5136:
5120:
5079:Divisor-related
5074:
5034:
4985:
4955:
4892:
4876:
4855:
4822:
4795:
4783:
4765:
4677:
4676:related numbers
4650:
4627:
4594:
4585:Perfect totient
4551:
4528:
4459:Highly abundant
4401:
4380:
4312:
4295:
4267:
4250:
4236:Stirling second
4142:
4119:
4080:
4062:
4019:
3968:
3905:
3866:Centered square
3834:
3817:
3779:
3764:
3731:
3716:
3668:
3667:defined numbers
3650:
3617:
3602:
3573:Double Mersenne
3559:
3540:
3462:
3448:
3446:natural numbers
3442:
3404:
3402:
3384:
3381:at Numberphile.
3368:Wayback Machine
3342:
3341:
3332:
3322:
3312:Springer-Verlag
3302:
3299:
3294:
3293:
3279:
3278:
3274:
3260:
3259:
3255:
3235:
3234:
3230:
3201:
3200:
3196:
3172:
3171:
3167:
3151:
3150:
3146:
3136:
3134:
3118:
3117:
3113:
3103:
3101:
3085:
3084:
3080:
3060:
3059:
3055:
3023:
3022:
3018:
3008:
3006:
3001:
3000:
2996:
2991:
2959:
2954:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2680:
2679:
2654:
2653:
2650:
2632:
2630:12-happy primes
2622:
2619:
2617:
2606:
2601:
2599:
2591:
2588:
2585:
2583:
2581:
2564:
2553:
2551:10-happy primes
2536:
2501:
2500:
2477:
2476:
2457:
2456:
2453:
2403:
2385:
2350:
2345:
2344:
2309:
2304:
2303:
2278:
2277:
2274:
2233:
2228:
2227:
2192:
2187:
2186:
2161:
2160:
2157:
2155:6-happy numbers
2121:
2116:
2115:
2084:
2079:
2078:
2053:
2052:
2049:
2047:4-happy numbers
2044:
2021:
2019:
2000:
1999:
1980:
1979:
1976:
1975:
1970:
1959:
1955:
1952:
1943:natural density
1939:
1907:
1906:
1887:
1886:
1865:
1860:
1859:
1838:
1833:
1832:
1813:
1812:
1793:
1792:
1773:
1772:
1742:
1729:
1713:
1694:
1666:
1628:
1627:
1599:
1586:
1573:
1557:
1538:
1510:
1472:
1471:
1443:
1430:
1417:
1404:
1388:
1369:
1341:
1336:
1335:
1302:
1289:
1276:
1248:
1210:
1209:
1181:
1168:
1140:
1102:
1101:
1073:
1060:
1032:
994:
993:
965:
952:
924:
919:
918:
893:
892:
865:
860:
859:
836:
835:
808:
803:
802:
779:
778:
777:represents the
741:
740:
688:
687:
668:
667:
648:
647:
628:
627:
602:
601:
563:
550:
521:
512:
506:
504:
481:
434:
429:
428:
403:
402:
399:
393:
344:
343:
320:
319:
289:
288:
250:
237:
232:
231:
204:
191:
186:
185:
158:
153:
152:
125:
112:
107:
106:
79:
66:
61:
60:
39:
28:
23:
22:
15:
12:
11:
5:
6657:
6655:
6647:
6646:
6641:
6631:
6630:
6624:
6623:
6618:
6615:
6614:
6612:
6611:
6606:
6601:
6596:
6591:
6586:
6581:
6576:
6571:
6566:
6561:
6556:
6551:
6546:
6541:
6536:
6531:
6526:
6521:
6516:
6511:
6506:
6501:
6496:
6491:
6486:
6481:
6476:
6471:
6466:
6461:
6456:
6451:
6446:
6441:
6436:
6431:
6426:
6421:
6416:
6411:
6406:
6401:
6396:
6391:
6386:
6381:
6376:
6371:
6366:
6361:
6356:
6351:
6346:
6341:
6336:
6331:
6326:
6321:
6316:
6310:
6308:
6304:
6303:
6301:
6300:
6295:
6290:
6285:
6280:
6278:Probable prime
6274:
6272:
6271:Related topics
6268:
6267:
6265:
6264:
6259:
6254:
6252:Sphenic number
6249:
6244:
6239:
6234:
6233:
6232:
6227:
6222:
6217:
6212:
6207:
6202:
6197:
6192:
6187:
6176:
6174:
6168:
6167:
6165:
6164:
6162:Gaussian prime
6159:
6153:
6151:
6145:
6144:
6142:
6141:
6140:
6130:
6125:
6123:
6119:
6118:
6116:
6115:
6091:
6087: + 1
6075:
6070:
6049:
6046:
6045:
6043:
6042:
6014:
5994:
5990: + 6
5978:
5974: + 4
5962:
5958: + 8
5938:
5934: + 6
5914:
5910: + 2
5897:
5895:
5883:
5879:
5878:
5876:
5875:
5870:
5865:
5860:
5855:
5850:
5845:
5840:
5835:
5830:
5825:
5820:
5815:
5810:
5805:
5797:
5792:
5786:
5784:
5777:
5776:
5774:
5773:
5768:
5763:
5758:
5753:
5748:
5743:
5738:
5733:
5728:
5723:
5718:
5713:
5708:
5703:
5698:
5693:
5688:
5677:
5675:
5671:
5670:
5668:
5667:
5662:
5657:
5652:
5647:
5641:
5639:
5635:
5634:
5632:
5631:
5614:
5610: â 1
5597:
5588:
5573:
5550:
5538:
5526:
5517:
5502:
5493:
5489: + 1
5480:
5472:
5465:
5457:
5450:
5438:
5426:
5418:
5409:
5400:
5390:
5388:
5384:
5383:
5377:
5375:
5374:
5367:
5360:
5352:
5343:
5342:
5340:
5339:
5328:
5325:
5324:
5321:
5320:
5318:
5317:
5311:
5308:
5307:
5301:
5294:
5293:
5290:
5289:
5287:
5286:
5281:
5275:
5272:
5271:
5265:
5258:
5257:
5254:
5253:
5251:
5250:
5248:Sorting number
5245:
5243:Pancake number
5239:
5236:
5235:
5229:
5222:
5221:
5218:
5217:
5215:
5214:
5209:
5203:
5200:
5199:
5193:
5186:
5185:
5182:
5181:
5179:
5178:
5173:
5168:
5162:
5159:
5158:
5155:Binary numbers
5153:
5146:
5145:
5142:
5141:
5138:
5137:
5135:
5134:
5128:
5126:
5122:
5121:
5119:
5118:
5113:
5108:
5103:
5098:
5093:
5088:
5082:
5080:
5076:
5075:
5073:
5072:
5067:
5062:
5057:
5052:
5046:
5044:
5036:
5035:
5033:
5032:
5027:
5022:
5017:
5012:
5007:
5002:
4996:
4994:
4987:
4986:
4984:
4983:
4982:
4981:
4970:
4968:
4965:P-adic numbers
4961:
4960:
4957:
4956:
4954:
4953:
4952:
4951:
4941:
4936:
4931:
4926:
4921:
4916:
4911:
4906:
4900:
4898:
4894:
4893:
4891:
4890:
4884:
4882:
4881:Coding-related
4878:
4877:
4875:
4874:
4869:
4863:
4861:
4857:
4856:
4854:
4853:
4848:
4843:
4838:
4832:
4830:
4821:
4820:
4819:
4818:
4816:Multiplicative
4813:
4802:
4800:
4785:
4784:
4780:Numeral system
4778:
4771:
4770:
4767:
4766:
4764:
4763:
4758:
4753:
4748:
4743:
4738:
4733:
4728:
4723:
4718:
4713:
4708:
4703:
4698:
4693:
4688:
4682:
4679:
4678:
4667:
4660:
4659:
4656:
4655:
4652:
4651:
4649:
4648:
4643:
4637:
4635:
4629:
4628:
4626:
4625:
4620:
4615:
4610:
4604:
4602:
4596:
4595:
4593:
4592:
4587:
4582:
4577:
4572:
4570:Highly totient
4567:
4561:
4559:
4553:
4552:
4550:
4549:
4544:
4538:
4536:
4530:
4529:
4527:
4526:
4521:
4516:
4511:
4506:
4501:
4496:
4491:
4486:
4481:
4476:
4471:
4466:
4461:
4456:
4451:
4446:
4441:
4436:
4431:
4426:
4424:Almost perfect
4421:
4415:
4413:
4403:
4402:
4393:
4386:
4385:
4382:
4381:
4379:
4378:
4373:
4368:
4363:
4358:
4353:
4348:
4343:
4338:
4333:
4328:
4323:
4317:
4314:
4313:
4308:
4301:
4300:
4297:
4296:
4294:
4293:
4288:
4283:
4278:
4272:
4269:
4268:
4263:
4256:
4255:
4252:
4251:
4249:
4248:
4243:
4238:
4233:
4231:Stirling first
4228:
4223:
4218:
4213:
4208:
4203:
4198:
4193:
4188:
4183:
4178:
4173:
4168:
4163:
4158:
4153:
4147:
4144:
4143:
4140:
4133:
4132:
4129:
4128:
4125:
4124:
4121:
4120:
4118:
4117:
4112:
4107:
4101:
4099:
4092:
4086:
4085:
4082:
4081:
4079:
4078:
4072:
4070:
4064:
4063:
4061:
4060:
4055:
4050:
4045:
4040:
4035:
4029:
4027:
4021:
4020:
4018:
4017:
4012:
4007:
4002:
3997:
3991:
3989:
3980:
3974:
3973:
3970:
3969:
3967:
3966:
3961:
3956:
3951:
3946:
3941:
3936:
3931:
3926:
3921:
3915:
3913:
3907:
3906:
3904:
3903:
3898:
3893:
3888:
3883:
3878:
3873:
3868:
3863:
3857:
3855:
3846:
3836:
3835:
3830:
3823:
3822:
3819:
3818:
3816:
3815:
3810:
3805:
3800:
3795:
3790:
3784:
3781:
3780:
3777:
3770:
3769:
3766:
3765:
3763:
3762:
3757:
3752:
3747:
3742:
3736:
3733:
3732:
3729:
3722:
3721:
3718:
3717:
3715:
3714:
3709:
3704:
3699:
3694:
3689:
3684:
3679:
3673:
3670:
3669:
3663:
3656:
3655:
3652:
3651:
3649:
3648:
3643:
3638:
3633:
3628:
3622:
3619:
3618:
3615:
3608:
3607:
3604:
3603:
3601:
3600:
3595:
3590:
3585:
3580:
3575:
3570:
3564:
3561:
3560:
3553:
3546:
3545:
3542:
3541:
3539:
3538:
3533:
3528:
3523:
3518:
3513:
3508:
3503:
3498:
3493:
3488:
3483:
3478:
3473:
3467:
3464:
3463:
3457:
3450:
3449:
3443:
3441:
3440:
3433:
3426:
3418:
3412:
3411:
3385:Symonds, Ria.
3382:
3376:
3370:
3358:
3347:"Happy Number"
3339:
3331:
3330:External links
3328:
3327:
3326:
3320:
3298:
3295:
3292:
3291:
3272:
3253:
3228:
3194:
3165:
3144:
3111:
3078:
3053:
3016:
2993:
2992:
2990:
2987:
2986:
2985:
2980:
2975:
2973:Harshad number
2970:
2965:
2958:
2955:
2718:
2693:
2690:
2687:
2667:
2664:
2661:
2649:
2646:
2645:
2644:
2631:
2628:
2614:Mersenne prime
2575:
2574:
2552:
2549:
2548:
2547:
2535:
2534:6-happy primes
2532:
2508:
2484:
2464:
2452:
2449:
2448:
2447:
2431:
2430:
2414:
2413:
2396:
2395:
2363:
2360:
2357:
2353:
2341:
2340:
2322:
2319:
2316:
2312:
2291:
2288:
2285:
2273:
2270:
2269:
2268:
2246:
2243:
2240:
2236:
2224:
2223:
2205:
2202:
2199:
2195:
2174:
2171:
2168:
2156:
2153:
2134:
2131:
2128:
2124:
2097:
2094:
2091:
2087:
2066:
2063:
2060:
2048:
2045:
2043:
2042:-happy numbers
2036:
2007:
1987:
1971:
1960:
1954:
1951:
1948:
1938:
1937:-happy numbers
1931:
1914:
1894:
1872:
1868:
1845:
1841:
1820:
1800:
1780:
1769:
1768:
1757:
1754:
1749:
1745:
1741:
1736:
1732:
1728:
1725:
1720:
1716:
1712:
1707:
1704:
1701:
1697:
1693:
1690:
1687:
1684:
1679:
1676:
1673:
1669:
1665:
1662:
1659:
1656:
1651:
1646:
1643:
1640:
1636:
1625:
1614:
1611:
1606:
1602:
1598:
1593:
1589:
1585:
1580:
1576:
1572:
1569:
1564:
1560:
1556:
1551:
1548:
1545:
1541:
1537:
1534:
1531:
1528:
1523:
1520:
1517:
1513:
1509:
1506:
1503:
1500:
1495:
1490:
1487:
1484:
1480:
1469:
1458:
1455:
1450:
1446:
1442:
1437:
1433:
1429:
1424:
1420:
1416:
1411:
1407:
1403:
1400:
1395:
1391:
1387:
1382:
1379:
1376:
1372:
1368:
1365:
1362:
1359:
1354:
1351:
1348:
1344:
1329:
1328:
1317:
1314:
1309:
1305:
1301:
1296:
1292:
1288:
1283:
1279:
1275:
1272:
1269:
1266:
1261:
1258:
1255:
1251:
1247:
1244:
1241:
1238:
1233:
1228:
1225:
1222:
1218:
1207:
1196:
1193:
1188:
1184:
1180:
1175:
1171:
1167:
1164:
1161:
1158:
1153:
1150:
1147:
1143:
1139:
1136:
1133:
1130:
1125:
1120:
1117:
1114:
1110:
1099:
1088:
1085:
1080:
1076:
1072:
1067:
1063:
1059:
1056:
1053:
1050:
1045:
1042:
1039:
1035:
1031:
1028:
1025:
1022:
1017:
1012:
1009:
1006:
1002:
991:
980:
977:
972:
968:
964:
959:
955:
951:
948:
945:
942:
937:
934:
931:
927:
900:
878:
875:
872:
868:
843:
821:
818:
815:
811:
786:
764:
759:
756:
753:
749:
728:
725:
722:
719:
716:
711:
706:
703:
700:
696:
675:
655:
635:
615:
612:
609:
598:
597:
583:
577:
570:
566:
557:
553:
548:
544:
541:
534:
531:
528:
524:
519:
515:
509:
501:
497:
493:
488:
484:
480:
475:
472:
469:
465:
461:
458:
455:
452:
447:
444:
441:
437:
410:
401:Formally, let
392:
389:
357:
354:
351:
327:
313:natural number
296:
265:
262:
257:
253:
249:
244:
240:
219:
216:
211:
207:
203:
198:
194:
173:
170:
165:
161:
140:
137:
132:
128:
124:
119:
115:
94:
91:
86:
82:
78:
73:
69:
32:Harshad number
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6656:
6645:
6642:
6640:
6637:
6636:
6634:
6621:
6616:
6610:
6607:
6605:
6602:
6600:
6597:
6595:
6592:
6590:
6587:
6585:
6582:
6580:
6577:
6575:
6572:
6570:
6567:
6565:
6562:
6560:
6557:
6555:
6552:
6550:
6547:
6545:
6542:
6540:
6537:
6535:
6532:
6530:
6527:
6525:
6522:
6520:
6517:
6515:
6512:
6510:
6507:
6505:
6502:
6500:
6497:
6495:
6492:
6490:
6487:
6485:
6482:
6480:
6477:
6475:
6472:
6470:
6467:
6465:
6462:
6460:
6457:
6455:
6452:
6450:
6447:
6445:
6442:
6440:
6437:
6435:
6432:
6430:
6427:
6425:
6422:
6420:
6417:
6415:
6412:
6410:
6407:
6405:
6402:
6400:
6397:
6395:
6392:
6390:
6387:
6385:
6382:
6380:
6377:
6375:
6372:
6370:
6367:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6340:
6337:
6335:
6332:
6330:
6327:
6325:
6322:
6320:
6317:
6315:
6312:
6311:
6309:
6305:
6299:
6296:
6294:
6291:
6289:
6288:Illegal prime
6286:
6284:
6281:
6279:
6276:
6275:
6273:
6269:
6263:
6260:
6258:
6255:
6253:
6250:
6248:
6245:
6243:
6240:
6238:
6235:
6231:
6228:
6226:
6223:
6221:
6218:
6216:
6213:
6211:
6208:
6206:
6203:
6201:
6198:
6196:
6193:
6191:
6188:
6186:
6183:
6182:
6181:
6178:
6177:
6175:
6173:
6169:
6163:
6160:
6158:
6155:
6154:
6152:
6150:
6146:
6139:
6136:
6135:
6134:
6133:Largest known
6131:
6129:
6126:
6124:
6120:
6114:
6110:
6106:
6102:
6098:
6092:
6090:
6086:
6082:
6076:
6074:
6071:
6069:
6065:
6061:
6057:
6051:
6050:
6041:
6038:
6035: +
6034:
6030:
6026:
6023: â
6022:
6015:
6013:
6009:
6005:
6002: +
6001:
5995:
5993:
5989:
5985:
5979:
5977:
5973:
5969:
5963:
5961:
5957:
5953:
5949:
5945:
5939:
5937:
5933:
5929:
5925:
5921:
5915:
5913:
5909:
5905:
5899:
5898:
5896:
5894:
5892:
5887:
5884:
5880:
5874:
5871:
5869:
5866:
5864:
5861:
5859:
5856:
5854:
5851:
5849:
5846:
5844:
5841:
5839:
5836:
5834:
5831:
5829:
5826:
5824:
5821:
5819:
5816:
5814:
5811:
5809:
5806:
5804:
5798:
5796:
5793:
5791:
5788:
5787:
5785:
5782:
5778:
5772:
5769:
5767:
5764:
5762:
5759:
5757:
5754:
5752:
5749:
5747:
5744:
5742:
5739:
5737:
5734:
5732:
5729:
5727:
5724:
5722:
5719:
5717:
5714:
5712:
5709:
5707:
5704:
5702:
5699:
5697:
5694:
5692:
5689:
5686:
5682:
5679:
5678:
5676:
5672:
5666:
5663:
5661:
5658:
5656:
5653:
5651:
5648:
5646:
5643:
5642:
5640:
5636:
5630:
5624:
5615:
5613:
5609:
5605:
5598:
5596:
5589:
5587:
5584:
5581: +
5580:
5574:
5572:
5569:
5566: â
5565:
5561:
5558: â
5557:
5551:
5549:
5545:
5539:
5537:
5533:
5527:
5525:
5518:
5516:
5513:
5510: +
5509:
5503:
5501:
5494:
5492:
5488:
5483:Pythagorean (
5481:
5479:
5475:
5466:
5464:
5460:
5451:
5449:
5445:
5439:
5437:
5433:
5427:
5425:
5419:
5417:
5410:
5408:
5401:
5399:
5392:
5391:
5389:
5385:
5380:
5373:
5368:
5366:
5361:
5359:
5354:
5353:
5350:
5338:
5334:
5330:
5329:
5326:
5316:
5313:
5312:
5309:
5304:
5299:
5295:
5285:
5282:
5280:
5277:
5276:
5273:
5268:
5263:
5259:
5249:
5246:
5244:
5241:
5240:
5237:
5232:
5227:
5223:
5213:
5210:
5208:
5205:
5204:
5201:
5197:
5191:
5187:
5177:
5174:
5172:
5169:
5167:
5164:
5163:
5160:
5156:
5151:
5147:
5133:
5130:
5129:
5127:
5123:
5117:
5114:
5112:
5109:
5107:
5106:Polydivisible
5104:
5102:
5099:
5097:
5094:
5092:
5089:
5087:
5084:
5083:
5081:
5077:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5048:
5047:
5045:
5042:
5037:
5031:
5028:
5026:
5023:
5021:
5018:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4997:
4995:
4992:
4988:
4980:
4977:
4976:
4975:
4972:
4971:
4969:
4966:
4962:
4950:
4947:
4946:
4945:
4942:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4905:
4902:
4901:
4899:
4895:
4889:
4886:
4885:
4883:
4879:
4873:
4870:
4868:
4865:
4864:
4862:
4860:Digit product
4858:
4852:
4849:
4847:
4844:
4842:
4839:
4837:
4834:
4833:
4831:
4829:
4825:
4817:
4814:
4812:
4809:
4808:
4807:
4804:
4803:
4801:
4799:
4794:
4790:
4786:
4781:
4776:
4772:
4762:
4759:
4757:
4754:
4752:
4749:
4747:
4744:
4742:
4739:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4702:
4699:
4697:
4696:ErdĆsâNicolas
4694:
4692:
4689:
4687:
4684:
4683:
4680:
4675:
4671:
4665:
4661:
4647:
4644:
4642:
4639:
4638:
4636:
4634:
4630:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4605:
4603:
4601:
4597:
4591:
4588:
4586:
4583:
4581:
4578:
4576:
4573:
4571:
4568:
4566:
4563:
4562:
4560:
4558:
4554:
4548:
4545:
4543:
4540:
4539:
4537:
4535:
4531:
4525:
4522:
4520:
4517:
4515:
4514:Superabundant
4512:
4510:
4507:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4470:
4467:
4465:
4462:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4430:
4427:
4425:
4422:
4420:
4417:
4416:
4414:
4412:
4408:
4404:
4400:
4396:
4391:
4387:
4377:
4374:
4372:
4369:
4367:
4364:
4362:
4359:
4357:
4354:
4352:
4349:
4347:
4344:
4342:
4339:
4337:
4334:
4332:
4329:
4327:
4324:
4322:
4319:
4318:
4315:
4311:
4306:
4302:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4274:
4273:
4270:
4266:
4261:
4257:
4247:
4244:
4242:
4239:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4184:
4182:
4179:
4177:
4174:
4172:
4169:
4167:
4164:
4162:
4159:
4157:
4154:
4152:
4149:
4148:
4145:
4138:
4134:
4116:
4113:
4111:
4108:
4106:
4103:
4102:
4100:
4096:
4093:
4091:
4090:4-dimensional
4087:
4077:
4074:
4073:
4071:
4069:
4065:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4034:
4031:
4030:
4028:
4026:
4022:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
4000:Centered cube
3998:
3996:
3993:
3992:
3990:
3988:
3984:
3981:
3979:
3978:3-dimensional
3975:
3965:
3962:
3960:
3957:
3955:
3952:
3950:
3947:
3945:
3942:
3940:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3920:
3917:
3916:
3914:
3912:
3908:
3902:
3899:
3897:
3894:
3892:
3889:
3887:
3884:
3882:
3879:
3877:
3874:
3872:
3869:
3867:
3864:
3862:
3859:
3858:
3856:
3854:
3850:
3847:
3845:
3844:2-dimensional
3841:
3837:
3833:
3828:
3824:
3814:
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3788:Nonhypotenuse
3786:
3785:
3782:
3775:
3771:
3761:
3758:
3756:
3753:
3751:
3748:
3746:
3743:
3741:
3738:
3737:
3734:
3727:
3723:
3713:
3710:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3674:
3671:
3666:
3661:
3657:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3629:
3627:
3624:
3623:
3620:
3613:
3609:
3599:
3596:
3594:
3591:
3589:
3586:
3584:
3581:
3579:
3576:
3574:
3571:
3569:
3566:
3565:
3562:
3557:
3551:
3547:
3537:
3534:
3532:
3529:
3527:
3526:Perfect power
3524:
3522:
3519:
3517:
3516:Seventh power
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3468:
3465:
3460:
3455:
3451:
3447:
3439:
3434:
3432:
3427:
3425:
3420:
3419:
3416:
3400:
3396:
3392:
3388:
3383:
3380:
3377:
3374:
3373:Happy Numbers
3371:
3369:
3365:
3362:
3359:
3354:
3353:
3348:
3345:
3340:
3338:
3334:
3333:
3329:
3323:
3321:0-387-20860-7
3317:
3313:
3309:
3305:
3301:
3300:
3296:
3287:
3283:
3276:
3273:
3268:
3264:
3257:
3254:
3249:
3248:
3242:
3238:
3232:
3229:
3225:
3219:
3215:
3211:
3210:
3205:
3198:
3195:
3190:
3189:
3183:
3181:
3175:
3169:
3166:
3160:
3155:
3148:
3145:
3132:
3131:
3125:
3121:
3115:
3112:
3099:
3098:
3092:
3088:
3082:
3079:
3074:
3073:
3067:
3063:
3057:
3054:
3049:
3045:
3040:
3035:
3031:
3027:
3020:
3017:
3004:
2998:
2995:
2988:
2984:
2981:
2979:
2976:
2974:
2971:
2969:
2966:
2964:
2961:
2960:
2956:
2716:
2714:
2709:
2707:
2691:
2688:
2685:
2665:
2662:
2659:
2647:
2641:
2640:
2639:
2637:
2629:
2627:
2615:
2610:
2580:
2572:
2567:
2562:
2561:
2560:
2558:
2550:
2545:
2544:
2543:
2541:
2533:
2531:
2529:
2525:
2520:
2506:
2498:
2482:
2462:
2450:
2444:
2443:
2442:
2440:
2435:
2428:
2427:
2426:
2424:
2420:
2411:
2406:
2401:
2400:
2399:
2393:
2388:
2382:
2381:
2380:
2377:
2361:
2358:
2355:
2351:
2338:
2337:
2336:
2320:
2317:
2314:
2310:
2289:
2286:
2283:
2271:
2265:
2264:
2263:
2260:
2244:
2241:
2238:
2234:
2221:
2220:
2219:
2203:
2200:
2197:
2193:
2172:
2169:
2166:
2154:
2152:
2150:
2132:
2129:
2126:
2122:
2113:
2095:
2092:
2089:
2085:
2064:
2061:
2058:
2046:
2041:
2037:
2035:
2033:
2029:
2005:
1985:
1974:
1968:
1964:
1949:
1947:
1944:
1936:
1932:
1930:
1928:
1912:
1892:
1870:
1866:
1843:
1839:
1818:
1798:
1778:
1755:
1752:
1747:
1743:
1739:
1734:
1730:
1726:
1718:
1714:
1705:
1702:
1699:
1695:
1691:
1685:
1677:
1674:
1671:
1667:
1663:
1657:
1649:
1644:
1641:
1638:
1634:
1626:
1612:
1609:
1604:
1600:
1596:
1591:
1587:
1583:
1578:
1574:
1570:
1562:
1558:
1549:
1546:
1543:
1539:
1535:
1529:
1521:
1518:
1515:
1511:
1507:
1501:
1493:
1488:
1485:
1482:
1478:
1470:
1456:
1453:
1448:
1444:
1440:
1435:
1431:
1427:
1422:
1418:
1414:
1409:
1405:
1401:
1393:
1389:
1380:
1377:
1374:
1370:
1366:
1360:
1352:
1349:
1346:
1342:
1334:
1333:
1332:
1315:
1312:
1307:
1303:
1299:
1294:
1290:
1286:
1281:
1277:
1273:
1267:
1259:
1256:
1253:
1249:
1245:
1239:
1231:
1226:
1223:
1220:
1216:
1208:
1194:
1191:
1186:
1182:
1178:
1173:
1169:
1165:
1159:
1151:
1148:
1145:
1141:
1137:
1131:
1123:
1118:
1115:
1112:
1108:
1100:
1086:
1083:
1078:
1074:
1070:
1065:
1061:
1057:
1051:
1043:
1040:
1037:
1033:
1029:
1023:
1015:
1010:
1007:
1004:
1000:
992:
978:
975:
970:
966:
962:
957:
953:
949:
943:
935:
932:
929:
925:
917:
916:
915:
912:
898:
891:, then it is
876:
873:
870:
866:
857:
841:
819:
816:
813:
809:
800:
784:
762:
757:
754:
751:
747:
726:
723:
717:
709:
704:
701:
698:
694:
673:
653:
633:
613:
610:
607:
581:
575:
568:
564:
555:
551:
542:
539:
532:
529:
526:
522:
513:
507:
495:
491:
486:
482:
473:
470:
467:
463:
459:
453:
445:
442:
439:
435:
427:
426:
425:
424:
408:
398:
390:
388:
386:
382:
378:
374:
369:
355:
352:
349:
341:
325:
318:
314:
310:
294:
285:
283:
279:
263:
260:
255:
251:
247:
242:
238:
217:
214:
209:
205:
201:
196:
192:
171:
168:
163:
159:
138:
135:
130:
126:
122:
117:
113:
92:
89:
84:
80:
76:
71:
67:
58:
54:
53:number theory
45:
41:
37:
33:
19:
6242:Almost prime
6200:EulerâJacobi
6108:
6104:
6100:
6096:
6094:Cunningham (
6084:
6080:
6063:
6059:
6055:
6036:
6032:
6028:
6024:
6020:
6019:consecutive
6007:
6003:
5999:
5987:
5983:
5971:
5967:
5955:
5951:
5947:
5943:
5941:Quadruplet (
5931:
5927:
5923:
5919:
5907:
5903:
5890:
5838:Full reptend
5696:Wolstenholme
5691:WallâSunâSun
5622:
5607:
5603:
5582:
5578:
5567:
5563:
5559:
5555:
5543:
5531:
5511:
5507:
5486:
5470:
5455:
5443:
5431:
5379:Prime number
5070:Transposable
4948:
4934:Narcissistic
4841:Digital root
4761:Super-Poulet
4721:JordanâPĂłlya
4670:prime factor
4575:Noncototient
4542:Almost prime
4524:Superperfect
4499:Refactorable
4494:Quasiperfect
4469:Hyperperfect
4310:Pseudoprimes
4281:WallâSunâSun
4216:Ordered Bell
4186:FussâCatalan
4098:non-centered
4048:Dodecahedral
4025:non-centered
3911:non-centered
3813:Wolstenholme
3558:× 2 ± 1
3555:
3554:Of the form
3521:Eighth power
3501:Fourth power
3403:. Retrieved
3399:the original
3390:
3350:
3307:
3304:Guy, Richard
3285:
3275:
3266:
3256:
3244:
3231:
3213:
3207:
3197:
3185:
3179:
3168:
3159:math/0607213
3147:
3135:. Retrieved
3127:
3114:
3102:. Retrieved
3094:
3081:
3069:
3056:
3029:
3025:
3019:
3009:16 September
3007:. Retrieved
3003:"Sad Number"
2997:
2978:Lucky number
2930:pdi_function
2906:seen_numbers
2900:seen_numbers
2864:seen_numbers
2723:pdi_function
2710:
2651:
2633:
2611:
2576:
2554:
2537:
2521:
2454:
2451:Happy primes
2438:
2436:
2432:
2422:
2418:
2415:
2397:
2378:
2342:
2275:
2261:
2225:
2158:
2050:
2039:
1977:
1940:
1934:
1926:
1770:
1330:
913:
599:
400:
370:
309:happy number
308:
286:
281:
277:
57:happy number
56:
50:
40:
35:
6225:SomerâLucas
6180:Pseudoprime
5818:Truncatable
5790:Palindromic
5674:By property
5453:Primorial (
5441:Factorial (
5091:Extravagant
5086:Equidigital
5041:permutation
5000:Palindromic
4974:Automorphic
4872:Sum-product
4851:Sum-product
4806:Persistence
4701:ErdĆsâWoods
4623:Untouchable
4504:Semiperfect
4454:Hemiperfect
4115:Tesseractic
4053:Icosahedral
4033:Tetrahedral
3964:Dodecagonal
3665:Recursively
3536:Prime power
3511:Sixth power
3506:Fifth power
3486:Power of 10
3444:Classes of
3395:Brady Haran
3391:Numberphile
2495:-happy and
1950:Happy bases
626:, a number
373:Reg Allenby
317:number base
315:in a given
18:Happy prime
6633:Categories
6262:Pernicious
6257:Interprime
6017:Balanced (
5808:Permutable
5783:-dependent
5600:Williams (
5496:Pierpont (
5421:Wagstaff
5403:Mersenne (
5387:By formula
5303:Graphemics
5176:Pernicious
5030:Undulating
5005:Pandigital
4979:Trimorphic
4580:Nontotient
4429:Arithmetic
4043:Octahedral
3944:Heptagonal
3934:Pentagonal
3919:Triangular
3760:SierpiĆski
3682:Jacobsthal
3481:Power of 3
3476:Power of 2
3297:Literature
2989:References
911:-unhappy.
686:such that
395:See also:
6298:Prime gap
6247:Semiprime
6210:Frobenius
5917:Triplet (
5716:Ramanujan
5711:Fortunate
5681:Wieferich
5645:Fibonacci
5576:Leyland (
5541:Woodall (
5520:Solinas (
5505:Quartan (
5060:Parasitic
4909:Factorion
4836:Digit sum
4828:Digit sum
4646:Fortunate
4633:Primorial
4547:Semiprime
4484:Practical
4449:Descartes
4444:Deficient
4434:Betrothed
4276:Wieferich
4105:Pentatope
4068:pyramidal
3959:Decagonal
3954:Nonagonal
3949:Octagonal
3939:Hexagonal
3798:Practical
3745:Congruent
3677:Fibonacci
3641:Loeschian
3352:MathWorld
3222:Cited in
3039:1110.3836
2038:Specific
1927:happiness
799:iteration
600:for base
540:−
500:⌋
492:
479:⌊
464:∑
6190:Elliptic
5965:Cousin (
5882:Patterns
5873:Tetradic
5868:Dihedral
5833:Primeval
5828:Delicate
5813:Circular
5800:Repunit
5591:Thabit (
5529:Cullen (
5468:Euclid (
5394:Fermat (
5132:Friedman
5065:Primeval
5010:Repdigit
4967:-related
4914:Kaprekar
4888:Meertens
4811:Additive
4798:dynamics
4706:Friendly
4618:Sociable
4608:Amicable
4419:Abundant
4399:dynamics
4221:Schröder
4211:Narayana
4181:Eulerian
4171:Delannoy
4166:Dedekind
3987:centered
3853:centered
3740:Amenable
3697:Narayana
3687:Leonardo
3583:Mersenne
3531:Powerful
3471:Achilles
3364:Archived
3306:(2004).
3032:(2): 2.
3026:Integers
2957:See also
2834:is_happy
2626:digits.
1885:in base
739:, where
387::§E34).
385:Guy 2004
6185:Catalan
6122:By size
5893:-tuples
5823:Minimal
5726:Regular
5617:Mills (
5553:Cuban (
5429:Proth (
5381:classes
5305:related
5269:related
5233:related
5231:Sorting
5116:Vampire
5101:Harshad
5043:related
5015:Repunit
4929:Lychrel
4904:Dudeney
4756:StĂžrmer
4751:Sphenic
4736:Regular
4674:divisor
4613:Perfect
4509:Sublime
4479:Perfect
4206:Motzkin
4161:Catalan
3702:Padovan
3636:Leyland
3631:Idoneal
3626:Hilbert
3598:Woodall
3405:2 April
3286:utm.edu
3267:utm.edu
3239:(ed.).
3176:(ed.).
3137:8 April
3122:(ed.).
3104:8 April
3089:(ed.).
3064:(ed.).
3044:Bibcode
2636:base 12
2569:in the
2566:A035497
2557:base 10
2408:in the
2405:A124095
2390:in the
2387:A007770
282:unhappy
6230:Strong
6220:Perrin
6205:Fermat
5981:Sexy (
5901:Twin (
5843:Unique
5771:Unique
5731:Strong
5721:Pillai
5701:Wilson
5665:Perrin
5171:Odious
5096:Frugal
5050:Cyclic
5039:Digit-
4746:Smooth
4731:Pronic
4691:Cyclic
4668:Other
4641:Euclid
4291:Wilson
4265:Primes
3924:Square
3793:Polite
3755:Riesel
3750:Knödel
3712:Perrin
3593:Thabit
3578:Fermat
3568:Cullen
3491:Square
3459:Powers
3318:
2945:number
2942:return
2936:number
2924:number
2918:number
2891:number
2879:number
2840:number
2825:return
2816:number
2810:number
2792:number
2768:number
2729:number
2713:Python
2540:base 6
2528:base 4
2524:base 2
2149:base 4
2032:base 4
2028:base 2
1967:base 4
1963:base 2
834:, and
105:, and
6215:Lucas
6195:Euler
5848:Happy
5795:Emirp
5761:Higgs
5756:Super
5736:Stern
5706:Lucky
5650:Lucas
5212:Prime
5207:Lucky
5196:sieve
5125:Other
5111:Smith
4991:Digit
4949:Happy
4924:Keith
4897:Other
4741:Rough
4711:Giuga
4176:Euler
4038:Cubic
3692:Lucas
3588:Proth
3154:arXiv
3034:arXiv
2876:while
2852:->
2828:total
2780:total
2765:while
2756:total
2582:10 +
2497:prime
1905:) is
311:is a
36:harĆa
6138:list
6073:Chen
5853:Self
5781:Base
5751:Good
5685:pair
5655:Pell
5606:â1)·
5166:Evil
4846:Self
4796:and
4686:Blum
4397:and
4201:Lobb
4156:Cake
4151:Bell
3901:Star
3808:Ulam
3707:Pell
3496:Cube
3407:2013
3316:ISBN
3245:The
3186:The
3139:2011
3128:The
3106:2011
3095:The
3070:The
3011:2009
2882:>
2855:bool
2822:base
2798:base
2771:>
2735:base
2577:The
2571:OEIS
2526:and
2410:OEIS
2392:OEIS
2276:For
2159:For
2114:for
2051:For
2030:and
2026:are
1965:and
1961:Are
1390:1335
797:-th
611:>
342:for
184:and
55:, a
6609:281
6604:277
6599:271
6594:269
6589:263
6584:257
6579:251
6574:241
6569:239
6564:233
6559:229
6554:227
6549:223
6544:211
6539:199
6534:197
6529:193
6524:191
6519:181
6514:179
6509:173
6504:167
6499:163
6494:157
6489:151
6484:149
6479:139
6474:137
6469:131
6464:127
6459:113
6454:109
6449:107
6444:103
6439:101
6099:, 2
6083:, 2
6004:a·n
5562:)/(
5284:Ban
4672:or
4191:Lah
2912:add
2894:not
2888:and
2870:set
2846:int
2831:def
2786:pow
2741:int
2720:def
2634:In
2623:064
2620:837
2602:007
2600:150
2596:+ 1
2589:247
2586:426
2555:In
2538:In
1658:347
1559:112
1502:347
1361:347
1268:100
1195:100
858:of
801:of
646:is
547:mod
518:mod
483:log
379:at
280:or
278:sad
51:In
6635::
6434:97
6429:89
6424:83
6419:79
6414:73
6409:71
6404:67
6399:61
6394:59
6389:53
6384:47
6379:43
6374:41
6369:37
6364:31
6359:29
6354:23
6349:19
6344:17
6339:13
6334:11
6031:,
6027:,
6006:,
5986:,
5970:,
5946:,
5922:,
5906:,
3393:.
3389:.
3349:.
3314:.
3284:.
3265:.
3243:.
3214:13
3212:.
3206:.
3184:.
3126:.
3093:.
3068:.
3042:.
3030:13
3028:.
2948:==
2897:in
2873:()
2819://
2783:+=
2750:):
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2708:.
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2618:12
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2573:).
2455:A
2412:).
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2290:10
2034:.
2024:10
1867:10
1831:,
1715:10
1530:44
1457:44
1260:10
1240:19
1227:10
1160:68
1152:10
1132:19
1119:10
1087:68
1052:82
1044:10
1024:19
1011:10
979:82
944:19
936:10
368:.
284:.
218:37
172:16
93:10
6329:7
6324:5
6319:3
6314:2
6113:)
6109:p
6105:p
6101:p
6097:p
6089:)
6085:p
6081:p
6068:)
6064:n
6060:n
6056:n
6040:)
6037:n
6033:p
6029:p
6025:n
6021:p
6012:)
6008:n
6000:p
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5988:p
5984:p
5976:)
5972:p
5968:p
5960:)
5956:p
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5944:p
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5932:p
5928:p
5924:p
5920:p
5912:)
5908:p
5904:p
5891:k
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5683:(
5629:)
5626:â
5623:A
5620:â
5612:)
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5595:)
5586:)
5583:y
5579:x
5571:)
5568:y
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5512:y
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5485:4
5478:)
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5458:n
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5444:n
5436:)
5432:k
5416:)
5407:)
5398:)
5371:e
5364:t
5357:v
3556:a
3437:e
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3423:v
3409:.
3355:.
3324:.
3288:.
3269:.
3226:.
3220:.
3180:n
3162:.
3156::
3141:.
3108:.
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3036::
3013:.
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2921:)
2915:(
2909:.
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2843::
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2807:)
2804:2
2801:,
2795:%
2789:(
2777::
2774:0
2762:0
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2744:=
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2732:,
2726:(
2689:=
2686:b
2666:2
2663:=
2660:p
2592:Ă
2584:7
2507:b
2483:b
2463:b
2439:n
2423:n
2419:n
2362:b
2359:,
2356:2
2352:F
2321:b
2318:,
2315:2
2311:F
2287:=
2284:b
2245:b
2242:,
2239:2
2235:F
2204:b
2201:,
2198:2
2194:F
2173:6
2170:=
2167:b
2133:b
2130:,
2127:2
2123:F
2096:b
2093:,
2090:2
2086:F
2065:4
2062:=
2059:b
2040:b
2022:Ă
2020:5
2006:b
1986:b
1958::
1935:b
1913:b
1893:b
1871:n
1858:(
1844:n
1840:b
1819:n
1799:b
1779:b
1756:1
1753:=
1748:2
1744:0
1740:+
1735:2
1731:1
1727:=
1724:)
1719:6
1711:(
1706:6
1703:,
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1696:F
1692:=
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1686:6
1683:(
1678:6
1675:,
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1668:F
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1661:)
1655:(
1650:3
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1639:2
1635:F
1613:6
1610:=
1605:2
1601:2
1597:+
1592:2
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1584:+
1579:2
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1571:=
1568:)
1563:6
1555:(
1550:6
1547:,
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1540:F
1536:=
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1527:(
1522:6
1519:,
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1512:F
1508:=
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1499:(
1494:2
1489:6
1486:,
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1479:F
1454:=
1449:2
1445:5
1441:+
1436:2
1432:3
1428:+
1423:2
1419:3
1415:+
1410:2
1406:1
1402:=
1399:)
1394:6
1386:(
1381:6
1378:,
1375:2
1371:F
1367:=
1364:)
1358:(
1353:6
1350:,
1347:2
1343:F
1316:1
1313:=
1308:2
1304:0
1300:+
1295:2
1291:0
1287:+
1282:2
1278:1
1274:=
1271:)
1265:(
1257:,
1254:2
1250:F
1246:=
1243:)
1237:(
1232:4
1224:,
1221:2
1217:F
1192:=
1187:2
1183:8
1179:+
1174:2
1170:6
1166:=
1163:)
1157:(
1149:,
1146:2
1142:F
1138:=
1135:)
1129:(
1124:3
1116:,
1113:2
1109:F
1084:=
1079:2
1075:2
1071:+
1066:2
1062:8
1058:=
1055:)
1049:(
1041:,
1038:2
1034:F
1030:=
1027:)
1021:(
1016:2
1008:,
1005:2
1001:F
976:=
971:2
967:9
963:+
958:2
954:1
950:=
947:)
941:(
933:,
930:2
926:F
899:b
877:b
874:,
871:2
867:F
842:b
820:b
817:,
814:2
810:F
785:j
763:j
758:b
755:,
752:2
748:F
727:1
724:=
721:)
718:n
715:(
710:j
705:b
702:,
699:2
695:F
674:j
654:b
634:n
614:1
608:b
596:.
582:p
576:)
569:i
565:b
556:i
552:b
543:n
533:1
530:+
527:i
523:b
514:n
508:(
496:n
487:b
474:0
471:=
468:i
460:=
457:)
454:n
451:(
446:b
443:,
440:p
436:F
409:n
356:2
353:=
350:p
326:b
307:-
295:b
264:4
261:=
256:2
252:0
248:+
243:2
239:2
215:=
210:2
206:6
202:+
197:2
193:1
169:=
164:2
160:4
139:1
136:=
131:2
127:0
123:+
118:2
114:1
90:=
85:2
81:3
77:+
72:2
68:1
20:)
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