2060:
4321:
111:. For example, 1,620 has prime factorization 2 Ă 3 Ă 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, respectively. All 5-smooth numbers are of the form 2 Ă 3 Ă 5, where
478:
64:. For example, a 7-smooth number is a number in which every prime factor is at most 7. Therefore, 49 = 7 and 15750 = 2 Ă 3 Ă 5 Ă 7 are both 7-smooth, while 11 and 702 = 2 Ă 3 Ă 13 are not 7-smooth. The term seems to have been coined by
651:
1430:
227:-smooth numbers, one ensures that the base cases of this recursion are small primes, for which efficient algorithms exist. (Large prime sizes require less-efficient algorithms such as
2423:
1075:
1159:
1120:
1328:
748:
368:
318:
1031:
973:
683:
924:
510:
849:
822:
775:
376:
944:
889:
869:
795:
533:
274:
126:
The 3-smooth numbers have also been called "harmonic numbers", although that name has other more widely used meanings. 5-smooth numbers are also called
2003:
1907:
1657:
1584:
1195:
2416:
568:
1558:
1161:). It is 16-powersmooth since its greatest prime factor power is 2 = 16. The number is also 17-powersmooth, 18-powersmooth, etc.
1217:
3223:
2409:
3218:
1862:
1377:
3233:
3213:
2049:
1209:
3926:
3506:
2246:
2059:
4345:
1599:
1360:
216:
3228:
4012:
1996:
89:
3328:
1237:
3678:
2997:
2790:
1080:
For example, 720 (2 Ă 3 Ă 5) is 5-smooth but not 5-powersmooth (because there are several prime powers greater than 5,
228:
3713:
3683:
3358:
3348:
2200:
3854:
3268:
3002:
2982:
3544:
3708:
4350:
3803:
3426:
3183:
2992:
2974:
2868:
2858:
2848:
2236:
1368:
266:
3688:
3931:
3476:
3097:
2883:
2878:
2873:
2863:
2840:
2221:
2916:
1989:
1198:), e.g. the 9-powersmooth numbers (also the 10-powersmooth numbers) are exactly the positive divisors of 2520.
141:
3173:
1600:"Python: Get the Hamming numbers upto a given numbers also check whether a given number is an Hamming number"
4042:
4007:
3793:
3703:
3577:
3552:
3461:
3451:
3063:
3045:
2965:
2375:
2241:
2165:
1456:
1441:
513:
4302:
3572:
3446:
3077:
2853:
2633:
2560:
2226:
2185:
251:
239:
212:
3557:
3411:
3338:
2493:
2155:
2024:
1181:
262:
4266:
3906:
1043:
4199:
4093:
4057:
3798:
3521:
3501:
3318:
2987:
2775:
2747:
2329:
2231:
1261:
1236:
increases, the performance of the algorithm or method in question degrades rapidly. For example, the
1125:
1086:
257:
Smooth numbers have a number of applications to cryptography. While most applications center around
3921:
3785:
3780:
3748:
3511:
3486:
3481:
3456:
3386:
3382:
3313:
3203:
3035:
2831:
2800:
2390:
2385:
2180:
2175:
2160:
2099:
1675:(1965), "Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)",
1480:
1253:
1245:
4320:
1822:
1311:
710:
4324:
4078:
4073:
3987:
3961:
3859:
3838:
3610:
3491:
3441:
3363:
3333:
3273:
3040:
3020:
2951:
2664:
2314:
2309:
2270:
2190:
2170:
1778:
1742:
1708:
1692:
1241:
3208:
338:
288:
250:), and the problem of generating these numbers efficiently has been used as a test problem for
4218:
4163:
4017:
3992:
3966:
3743:
3421:
3416:
3343:
3323:
3308:
3030:
3012:
2931:
2921:
2906:
2684:
2669:
2350:
2290:
1888:
1858:
1554:
1280:
1009:
949:
659:
894:
486:
473:{\displaystyle \Psi (x,B)\sim {\frac {1}{\pi (B)!}}\prod _{p\leq B}{\frac {\log x}{\log p}}.}
4254:
4047:
3633:
3605:
3595:
3587:
3471:
3436:
3431:
3398:
3092:
3055:
2946:
2941:
2936:
2926:
2898:
2785:
2737:
2732:
2689:
2628:
2380:
2355:
2275:
2261:
2195:
2079:
2039:
1684:
1546:
686:
270:
180:
1704:
827:
800:
753:
4230:
4119:
4052:
3978:
3901:
3875:
3693:
3406:
3263:
3198:
3168:
3158:
3153:
2819:
2727:
2674:
2518:
2458:
2365:
2360:
2285:
2279:
2216:
2114:
2104:
2034:
1700:
1623:
1364:
77:
65:
1746:
4235:
4103:
4088:
3952:
3916:
3891:
3767:
3738:
3723:
3600:
3496:
3466:
3193:
3148:
3025:
2623:
2618:
2613:
2585:
2570:
2483:
2468:
2446:
2433:
2370:
2324:
2150:
2134:
2124:
2094:
1530:
1461:
929:
874:
854:
780:
518:
235:
188:
127:
1981:
4339:
4158:
4142:
4083:
4037:
3718:
3628:
3353:
2911:
2780:
2742:
2699:
2580:
2565:
2555:
2513:
2503:
2478:
2319:
2109:
2089:
1871:
1712:
258:
247:
57:
31:
4194:
4183:
4098:
3936:
3911:
3828:
3728:
3698:
3673:
3657:
3562:
3529:
3278:
3252:
3163:
3102:
2679:
2575:
2508:
2488:
2463:
2334:
2251:
2129:
2074:
2044:
1867:
1451:
1446:
1372:
243:
176:
104:
73:
69:
1891:
1506:
273:
hash function is another example of a constructive use of smoothness to obtain a
4153:
4028:
3833:
3297:
3188:
3143:
3138:
2888:
2795:
2694:
2523:
2498:
2473:
1806:
Kim, Taechan; Tibouchi, Mehdi (2015). "Invalid Curve
Attacks in a GLS Setting".
1647:
1574:
1550:
1356:
4290:
4271:
3567:
3178:
697:
17:
2401:
72:, which relies on factorization of integers. 2-smooth numbers are simply the
3896:
3823:
3815:
3620:
3534:
2652:
2084:
1896:
1337:= {3, 5}, as 12 contains the factor 4 = 2, and neither 4 nor 2 are in
646:{\displaystyle \Psi (x,y)=x\cdot \rho (u)+O\left({\frac {x}{\log y}}\right)}
1832:. Blacksburg, Virginia: Virginia Polytechnic Institute and State University
1220:. Such applications are often said to work with "smooth numbers," with no
3997:
1351:
does not have to be a set of prime factors, but it is typically a proper
219:), which operates by recursively breaking down a problem of a given size
1294:. For example, since 12 = 4 Ă 3, 12 is smooth over the sets
4002:
3661:
2029:
1696:
92:
53:
1767:
1533:; Reyneri, J. M. (1983). "Fast Computation of Discrete Logarithms in
1352:
1688:
1208:-powersmooth numbers have applications in number theory, such as in
1754:, Report EWD792. Originally a privately circulated handwritten note
2300:
1783:
1672:
1425:{\displaystyle \phi :\mathbb {Z} \to \mathbb {Z} /n\mathbb {Z} }
4288:
4252:
4216:
4180:
4140:
3765:
3654:
3380:
3295:
3250:
3127:
2817:
2764:
2716:
2650:
2602:
2540:
2444:
2405:
1985:
1180:-powersmooth numbers are exactly the positive divisors of âthe
1726:
Longuet-Higgins, H. C. (1962), "Letter to a musical friend",
211:
An important practical application of smooth numbers is the
1976:
1970:
1964:
1958:
1951:
1944:
1937:
1930:
1923:
1651:
1578:
1190:
155:-smooth number. If the largest prime factor of a number is
1768:"Divisibility, Smoothness and Cryptographic Applications"
1855:
Introduction to analytic and probabilistic number theory
151:
itself is not required to appear among the factors of a
926:-smooth part of a random integer less than or equal to
1873:
Smooth numbers: Computational number theory and beyond
1766:
Naccache, David; Shparlinski, Igor (17 October 2008).
1380:
1314:
1128:
1089:
1046:
1012:
952:
932:
897:
877:
857:
830:
803:
783:
756:
713:
662:
571:
521:
489:
379:
341:
291:
132:
or
Hamming numbers; 7-smooth numbers are also called
4112:
4066:
4026:
3977:
3951:
3884:
3868:
3847:
3814:
3779:
3619:
3586:
3543:
3520:
3397:
3085:
3076:
3054:
3011:
2973:
2964:
2897:
2839:
2830:
2343:
2299:
2260:
2209:
2143:
2067:
2017:
1823:"An Introduction to the General Number Field Sieve"
1224:specified; this means the numbers involved must be
1424:
1371:uses to build its notion of smoothness, under the
1322:
1153:
1114:
1069:
1025:
967:
938:
918:
883:
863:
843:
816:
789:
769:
742:
677:
645:
527:
504:
472:
362:
312:
140:, although this conflicts with another meaning of
1543:Advances in Cryptology â Proceedings of Crypto 82
335:is fixed and small, there is a good estimate for
1228:-powersmooth, for some unspecified small number
223:into problems the size of its factors. By using
1652:"Sequence A002473 (7-smooth numbers)"
1579:"Sequence A003586 (3-smooth numbers)"
1330:, however it would not be smooth over the set
2417:
1997:
68:. Smooth numbers are especially important in
8:
1290:where the factors are powers of elements in
199:is the largest prime less than or equal to
76:, while 5-smooth numbers are also known as
4285:
4249:
4213:
4177:
4137:
3811:
3776:
3762:
3651:
3394:
3377:
3292:
3247:
3124:
3082:
2970:
2836:
2827:
2814:
2761:
2718:Possessing a specific set of other numbers
2713:
2647:
2599:
2537:
2441:
2424:
2410:
2402:
2004:
1990:
1982:
1168:-smooth numbers, for any positive integer
514:the number of primes less than or equal to
1908:On-Line Encyclopedia of Integer Sequences
1782:
1658:On-Line Encyclopedia of Integer Sequences
1585:On-Line Encyclopedia of Integer Sequences
1418:
1417:
1409:
1405:
1404:
1388:
1387:
1379:
1316:
1315:
1313:
1133:
1127:
1094:
1088:
1066:
1051:
1045:
1017:
1011:
951:
931:
906:
902:
896:
876:
856:
835:
829:
808:
802:
782:
761:
755:
734:
724:
712:
661:
621:
570:
520:
488:
441:
429:
401:
378:
340:
290:
946:is known to decay much more slowly than
1472:
324:-smooth integers less than or equal to
27:Integer having only small prime factors
1481:"P-Smooth Numbers or P-friable Number"
7:
1821:Briggs, Matthew E. (17 April 1998).
2012:Divisibility-based sets of integers
1286:if there exists a factorization of
1176:-powersmooth numbers, in fact, the
183:are permitted as well. A number is
572:
380:
342:
292:
25:
2050:Fundamental theorem of arithmetic
1070:{\displaystyle p^{\nu }\leq n.\,}
4319:
3927:Perfect digit-to-digit invariant
2058:
824:is not (or is equal to 1), then
538:Otherwise, define the parameter
1154:{\displaystyle 2^{4}=16\nleq 5}
1876:, Proc. of MSRI workshop, 2008
1401:
1398:
1392:
1115:{\displaystyle 3^{2}=9\nleq 5}
962:
956:
672:
666:
608:
602:
587:
575:
499:
493:
416:
410:
395:
383:
357:
345:
307:
295:
215:(FFT) algorithms (such as the
60:are all less than or equal to
1:
2766:Expressible via specific sums
1355:of the primes as seen in the
1172:there are only finitely many
242:. They are also important in
1677:Journal of Cuneiform Studies
1361:Dixon's factorization method
1323:{\displaystyle \mathbb {Z} }
1279:is said to be smooth over a
743:{\displaystyle n=n_{1}n_{2}}
700:natural numbers will not be
3855:Multiplicative digital root
1624:"Problem H: Humble Numbers"
1551:10.1007/978-1-4757-0602-4_1
1367:. Likewise, it is what the
891:. The relative size of the
123:are non-negative integers.
4367:
1914:-smooth numbers for small
1748:Hamming's exercise in SASL
1648:Sloane, N. J. A.
1575:Sloane, N. J. A.
1369:general number field sieve
1214: â 1 algorithm
363:{\displaystyle \Psi (x,B)}
328:(the de Bruijn function).
313:{\displaystyle \Psi (x,y)}
267:General number field sieve
217:CooleyâTukey FFT algorithm
4315:
4298:
4284:
4262:
4248:
4226:
4212:
4190:
4176:
4149:
4136:
3932:Perfect digital invariant
3775:
3761:
3669:
3650:
3507:Superior highly composite
3393:
3376:
3304:
3291:
3259:
3246:
3134:
3123:
2826:
2813:
2771:
2760:
2723:
2712:
2660:
2646:
2609:
2598:
2551:
2536:
2454:
2440:
2247:Superior highly composite
2056:
546: = log
265:algorithms, for example:
229:Bluestein's FFT algorithm
3545:Euler's totient function
3329:EulerâJacobi pseudoprime
2604:Other polynomial numbers
2144:Constrained divisor sums
1238:PohligâHellman algorithm
1026:{\displaystyle p^{\nu }}
968:{\displaystyle \rho (u)}
678:{\displaystyle \rho (u)}
331:If the smoothness bound
261:(e.g. the fastest known
142:highly composite numbers
3359:SomerâLucas pseudoprime
3349:LucasâCarmichael number
3184:Lazy caterer's sequence
1442:Highly composite number
919:{\displaystyle x^{1/u}}
550: / log
505:{\displaystyle \pi (B)}
238:play a special role in
136:, and sometimes called
4346:Analytic number theory
3234:WedderburnâEtherington
2634:Lucky numbers of Euler
1426:
1324:
1244:has a running time of
1155:
1116:
1071:
1027:
969:
940:
920:
885:
865:
845:
818:
791:
771:
744:
679:
647:
529:
506:
474:
364:
314:
275:provably secure design
252:functional programming
240:Babylonian mathematics
213:fast Fourier transform
3522:Prime omega functions
3339:Frobenius pseudoprime
3129:Combinatorial numbers
2998:Centered dodecahedral
2791:Primary pseudoperfect
2025:Integer factorization
1511:mathworld.wolfram.com
1427:
1325:
1182:least common multiple
1156:
1117:
1072:
1028:
970:
941:
921:
886:
866:
846:
844:{\displaystyle n_{1}}
819:
817:{\displaystyle n_{2}}
792:
772:
770:{\displaystyle n_{1}}
745:
680:
648:
530:
507:
475:
365:
320:denote the number of
315:
263:integer factorization
3981:-composition related
3781:Arithmetic functions
3383:Arithmetic functions
3319:Elliptic pseudoprime
3003:Centered icosahedral
2983:Centered tetrahedral
1468:Notes and references
1378:
1312:
1308:= {2, 3}, and
1126:
1087:
1044:
1010:
950:
930:
895:
875:
855:
828:
801:
781:
754:
711:
660:
569:
519:
487:
377:
339:
289:
171:. In many scenarios
3907:Kaprekar's constant
3427:Colossally abundant
3314:Catalan pseudoprime
3214:SchröderâHipparchus
2993:Centered octahedral
2869:Centered heptagonal
2859:Centered pentagonal
2849:Centered triangular
2449:and related numbers
2237:Colossally abundant
2068:Factorization forms
1975:23-smooth numbers:
1969:19-smooth numbers:
1963:17-smooth numbers:
1957:13-smooth numbers:
1950:11-smooth numbers:
1743:Dijkstra, Edsger W.
1628:www.eecs.qmul.ac.uk
1505:Weisstein, Eric W.
1242:discrete logarithms
979:Powersmooth numbers
159:then the number is
4325:Mathematics portal
4267:Aronson's sequence
4013:SmarandacheâWellin
3770:-dependent numbers
3477:Primitive abundant
3364:Strong pseudoprime
3354:Perrin pseudoprime
3334:Fermat pseudoprime
3274:Wolstenholme prime
3098:Squared triangular
2884:Centered decagonal
2879:Centered nonagonal
2874:Centered octagonal
2864:Centered hexagonal
2222:Primitive abundant
2210:With many divisors
1943:7-smooth numbers:
1936:5-smooth numbers:
1929:3-smooth numbers:
1922:2-smooth numbers:
1889:Weisstein, Eric W.
1661:. OEIS Foundation.
1588:. OEIS Foundation.
1422:
1320:
1268:Smooth over a set
1151:
1112:
1067:
1023:
965:
936:
916:
881:
861:
841:
814:
787:
767:
740:
675:
643:
525:
502:
470:
440:
360:
310:
4351:Integer sequences
4333:
4332:
4311:
4310:
4280:
4279:
4244:
4243:
4208:
4207:
4172:
4171:
4132:
4131:
4128:
4127:
3947:
3946:
3757:
3756:
3646:
3645:
3642:
3641:
3588:Aliquot sequences
3399:Divisor functions
3372:
3371:
3344:Lucas pseudoprime
3324:Euler pseudoprime
3309:Carmichael number
3287:
3286:
3242:
3241:
3119:
3118:
3115:
3114:
3111:
3110:
3072:
3071:
2960:
2959:
2917:Square triangular
2809:
2808:
2756:
2755:
2708:
2707:
2642:
2641:
2594:
2593:
2532:
2531:
2399:
2398:
1730:(August): 244â248
1560:978-1-4757-0604-8
1545:. pp. 3â13.
1457:StĂžrmer's theorem
939:{\displaystyle x}
884:{\displaystyle n}
864:{\displaystyle B}
790:{\displaystyle B}
637:
528:{\displaystyle B}
465:
425:
423:
181:composite numbers
107:are greater than
16:(Redirected from
4358:
4323:
4286:
4255:Natural language
4250:
4214:
4182:Generated via a
4178:
4138:
4043:Digit-reassembly
4008:Self-descriptive
3812:
3777:
3763:
3714:LucasâCarmichael
3704:Harmonic divisor
3652:
3578:Sparsely totient
3553:Highly cototient
3462:Multiply perfect
3452:Highly composite
3395:
3378:
3293:
3248:
3229:Telephone number
3125:
3083:
3064:Square pyramidal
3046:Stella octangula
2971:
2837:
2828:
2820:Figurate numbers
2815:
2762:
2714:
2648:
2600:
2538:
2442:
2426:
2419:
2412:
2403:
2376:Harmonic divisor
2262:Aliquot sequence
2242:Highly composite
2166:Multiply perfect
2062:
2040:Divisor function
2006:
1999:
1992:
1983:
1902:
1901:
1842:
1841:
1839:
1837:
1827:
1818:
1812:
1811:
1803:
1797:
1795:
1793:
1791:
1786:
1772:
1763:
1757:
1755:
1753:
1739:
1733:
1731:
1723:
1717:
1715:
1669:
1663:
1662:
1644:
1638:
1637:
1635:
1634:
1620:
1614:
1613:
1611:
1610:
1596:
1590:
1589:
1571:
1565:
1564:
1527:
1521:
1520:
1518:
1517:
1502:
1496:
1495:
1493:
1492:
1477:
1431:
1429:
1428:
1423:
1421:
1413:
1408:
1391:
1329:
1327:
1326:
1321:
1319:
1301:= {4, 3},
1193:
1160:
1158:
1157:
1152:
1138:
1137:
1121:
1119:
1118:
1113:
1099:
1098:
1076:
1074:
1073:
1068:
1056:
1055:
1032:
1030:
1029:
1024:
1022:
1021:
974:
972:
971:
966:
945:
943:
942:
937:
925:
923:
922:
917:
915:
914:
910:
890:
888:
887:
882:
871:-smooth part of
870:
868:
867:
862:
850:
848:
847:
842:
840:
839:
823:
821:
820:
815:
813:
812:
796:
794:
793:
788:
776:
774:
773:
768:
766:
765:
749:
747:
746:
741:
739:
738:
729:
728:
687:Dickman function
684:
682:
681:
676:
652:
650:
649:
644:
642:
638:
636:
622:
534:
532:
531:
526:
511:
509:
508:
503:
479:
477:
476:
471:
466:
464:
453:
442:
439:
424:
422:
402:
369:
367:
366:
361:
319:
317:
316:
311:
269:algorithm), the
163:-smooth for any
147:Here, note that
138:highly composite
21:
4366:
4365:
4361:
4360:
4359:
4357:
4356:
4355:
4336:
4335:
4334:
4329:
4307:
4303:Strobogrammatic
4294:
4276:
4258:
4240:
4222:
4204:
4186:
4168:
4145:
4124:
4108:
4067:Divisor-related
4062:
4022:
3973:
3943:
3880:
3864:
3843:
3810:
3783:
3771:
3753:
3665:
3664:related numbers
3638:
3615:
3582:
3573:Perfect totient
3539:
3516:
3447:Highly abundant
3389:
3368:
3300:
3283:
3255:
3238:
3224:Stirling second
3130:
3107:
3068:
3050:
3007:
2956:
2893:
2854:Centered square
2822:
2805:
2767:
2752:
2719:
2704:
2656:
2655:defined numbers
2638:
2605:
2590:
2561:Double Mersenne
2547:
2528:
2450:
2436:
2434:natural numbers
2430:
2400:
2395:
2339:
2295:
2256:
2227:Highly abundant
2205:
2186:Unitary perfect
2139:
2063:
2054:
2035:Unitary divisor
2013:
2010:
1892:"Smooth Number"
1887:
1886:
1883:
1850:
1845:
1835:
1833:
1825:
1820:
1819:
1815:
1805:
1804:
1800:
1789:
1787:
1775:eprint.iacr.org
1770:
1765:
1764:
1760:
1751:
1741:
1740:
1736:
1725:
1724:
1720:
1689:10.2307/1359089
1671:
1670:
1666:
1646:
1645:
1641:
1632:
1630:
1622:
1621:
1617:
1608:
1606:
1598:
1597:
1593:
1573:
1572:
1568:
1561:
1529:
1528:
1524:
1515:
1513:
1507:"Smooth Number"
1504:
1503:
1499:
1490:
1488:
1479:
1478:
1474:
1470:
1438:
1376:
1375:
1365:quadratic sieve
1343:
1336:
1310:
1309:
1307:
1300:
1273:
1189:
1184:of 1, 2, 3, âŠ,
1129:
1124:
1123:
1090:
1085:
1084:
1047:
1042:
1041:
1013:
1008:
1007:
1003:) if all prime
981:
948:
947:
928:
927:
898:
893:
892:
873:
872:
853:
852:
831:
826:
825:
804:
799:
798:
779:
778:
757:
752:
751:
730:
720:
709:
708:
658:
657:
626:
617:
567:
566:
558: =
517:
516:
485:
484:
454:
443:
406:
375:
374:
337:
336:
287:
286:
283:
236:regular numbers
209:
202:
198:
195:-smooth, where
194:
186:
174:
170:
166:
162:
158:
154:
150:
129:regular numbers
110:
103:if none of its
98:
86:
78:regular numbers
66:Leonard Adleman
28:
23:
22:
15:
12:
11:
5:
4364:
4362:
4354:
4353:
4348:
4338:
4337:
4331:
4330:
4328:
4327:
4316:
4313:
4312:
4309:
4308:
4306:
4305:
4299:
4296:
4295:
4289:
4282:
4281:
4278:
4277:
4275:
4274:
4269:
4263:
4260:
4259:
4253:
4246:
4245:
4242:
4241:
4239:
4238:
4236:Sorting number
4233:
4231:Pancake number
4227:
4224:
4223:
4217:
4210:
4209:
4206:
4205:
4203:
4202:
4197:
4191:
4188:
4187:
4181:
4174:
4173:
4170:
4169:
4167:
4166:
4161:
4156:
4150:
4147:
4146:
4143:Binary numbers
4141:
4134:
4133:
4130:
4129:
4126:
4125:
4123:
4122:
4116:
4114:
4110:
4109:
4107:
4106:
4101:
4096:
4091:
4086:
4081:
4076:
4070:
4068:
4064:
4063:
4061:
4060:
4055:
4050:
4045:
4040:
4034:
4032:
4024:
4023:
4021:
4020:
4015:
4010:
4005:
4000:
3995:
3990:
3984:
3982:
3975:
3974:
3972:
3971:
3970:
3969:
3958:
3956:
3953:P-adic numbers
3949:
3948:
3945:
3944:
3942:
3941:
3940:
3939:
3929:
3924:
3919:
3914:
3909:
3904:
3899:
3894:
3888:
3886:
3882:
3881:
3879:
3878:
3872:
3870:
3869:Coding-related
3866:
3865:
3863:
3862:
3857:
3851:
3849:
3845:
3844:
3842:
3841:
3836:
3831:
3826:
3820:
3818:
3809:
3808:
3807:
3806:
3804:Multiplicative
3801:
3790:
3788:
3773:
3772:
3768:Numeral system
3766:
3759:
3758:
3755:
3754:
3752:
3751:
3746:
3741:
3736:
3731:
3726:
3721:
3716:
3711:
3706:
3701:
3696:
3691:
3686:
3681:
3676:
3670:
3667:
3666:
3655:
3648:
3647:
3644:
3643:
3640:
3639:
3637:
3636:
3631:
3625:
3623:
3617:
3616:
3614:
3613:
3608:
3603:
3598:
3592:
3590:
3584:
3583:
3581:
3580:
3575:
3570:
3565:
3560:
3558:Highly totient
3555:
3549:
3547:
3541:
3540:
3538:
3537:
3532:
3526:
3524:
3518:
3517:
3515:
3514:
3509:
3504:
3499:
3494:
3489:
3484:
3479:
3474:
3469:
3464:
3459:
3454:
3449:
3444:
3439:
3434:
3429:
3424:
3419:
3414:
3412:Almost perfect
3409:
3403:
3401:
3391:
3390:
3381:
3374:
3373:
3370:
3369:
3367:
3366:
3361:
3356:
3351:
3346:
3341:
3336:
3331:
3326:
3321:
3316:
3311:
3305:
3302:
3301:
3296:
3289:
3288:
3285:
3284:
3282:
3281:
3276:
3271:
3266:
3260:
3257:
3256:
3251:
3244:
3243:
3240:
3239:
3237:
3236:
3231:
3226:
3221:
3219:Stirling first
3216:
3211:
3206:
3201:
3196:
3191:
3186:
3181:
3176:
3171:
3166:
3161:
3156:
3151:
3146:
3141:
3135:
3132:
3131:
3128:
3121:
3120:
3117:
3116:
3113:
3112:
3109:
3108:
3106:
3105:
3100:
3095:
3089:
3087:
3080:
3074:
3073:
3070:
3069:
3067:
3066:
3060:
3058:
3052:
3051:
3049:
3048:
3043:
3038:
3033:
3028:
3023:
3017:
3015:
3009:
3008:
3006:
3005:
3000:
2995:
2990:
2985:
2979:
2977:
2968:
2962:
2961:
2958:
2957:
2955:
2954:
2949:
2944:
2939:
2934:
2929:
2924:
2919:
2914:
2909:
2903:
2901:
2895:
2894:
2892:
2891:
2886:
2881:
2876:
2871:
2866:
2861:
2856:
2851:
2845:
2843:
2834:
2824:
2823:
2818:
2811:
2810:
2807:
2806:
2804:
2803:
2798:
2793:
2788:
2783:
2778:
2772:
2769:
2768:
2765:
2758:
2757:
2754:
2753:
2751:
2750:
2745:
2740:
2735:
2730:
2724:
2721:
2720:
2717:
2710:
2709:
2706:
2705:
2703:
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2661:
2658:
2657:
2651:
2644:
2643:
2640:
2639:
2637:
2636:
2631:
2626:
2621:
2616:
2610:
2607:
2606:
2603:
2596:
2595:
2592:
2591:
2589:
2588:
2583:
2578:
2573:
2568:
2563:
2558:
2552:
2549:
2548:
2541:
2534:
2533:
2530:
2529:
2527:
2526:
2521:
2516:
2511:
2506:
2501:
2496:
2491:
2486:
2481:
2476:
2471:
2466:
2461:
2455:
2452:
2451:
2445:
2438:
2437:
2431:
2429:
2428:
2421:
2414:
2406:
2397:
2396:
2394:
2393:
2388:
2383:
2378:
2373:
2368:
2363:
2358:
2353:
2347:
2345:
2341:
2340:
2338:
2337:
2332:
2327:
2322:
2317:
2312:
2306:
2304:
2297:
2296:
2294:
2293:
2288:
2283:
2273:
2267:
2265:
2258:
2257:
2255:
2254:
2249:
2244:
2239:
2234:
2229:
2224:
2219:
2213:
2211:
2207:
2206:
2204:
2203:
2198:
2193:
2188:
2183:
2178:
2173:
2168:
2163:
2158:
2156:Almost perfect
2153:
2147:
2145:
2141:
2140:
2138:
2137:
2132:
2127:
2122:
2117:
2112:
2107:
2102:
2097:
2092:
2087:
2082:
2077:
2071:
2069:
2065:
2064:
2057:
2055:
2053:
2052:
2047:
2042:
2037:
2032:
2027:
2021:
2019:
2015:
2014:
2011:
2009:
2008:
2001:
1994:
1986:
1980:
1979:
1973:
1967:
1961:
1955:
1948:
1941:
1934:
1927:
1904:
1903:
1882:
1881:External links
1879:
1878:
1877:
1865:
1863:978-0821898543
1857:, (AMS, 2015)
1853:G. Tenenbaum,
1849:
1846:
1844:
1843:
1813:
1798:
1758:
1734:
1718:
1664:
1639:
1615:
1591:
1566:
1559:
1531:Hellman, M. E.
1522:
1497:
1471:
1469:
1466:
1465:
1464:
1462:Unusual number
1459:
1454:
1449:
1444:
1437:
1434:
1420:
1416:
1412:
1407:
1403:
1400:
1397:
1394:
1390:
1386:
1383:
1341:
1334:
1318:
1305:
1298:
1272:
1266:
1240:for computing
1150:
1147:
1144:
1141:
1136:
1132:
1111:
1108:
1105:
1102:
1097:
1093:
1078:
1077:
1065:
1062:
1059:
1054:
1050:
1020:
1016:
980:
977:
964:
961:
958:
955:
935:
913:
909:
905:
901:
880:
860:
851:is called the
838:
834:
811:
807:
786:
764:
760:
737:
733:
727:
723:
719:
716:
674:
671:
668:
665:
654:
653:
641:
635:
632:
629:
625:
620:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
524:
501:
498:
495:
492:
481:
480:
469:
463:
460:
457:
452:
449:
446:
438:
435:
432:
428:
421:
418:
415:
412:
409:
405:
400:
397:
394:
391:
388:
385:
382:
359:
356:
353:
350:
347:
344:
309:
306:
303:
300:
297:
294:
282:
279:
208:
205:
200:
196:
192:
189:if and only if
184:
172:
168:
164:
160:
156:
152:
148:
134:humble numbers
108:
96:
85:
82:
26:
24:
18:Friable number
14:
13:
10:
9:
6:
4:
3:
2:
4363:
4352:
4349:
4347:
4344:
4343:
4341:
4326:
4322:
4318:
4317:
4314:
4304:
4301:
4300:
4297:
4292:
4287:
4283:
4273:
4270:
4268:
4265:
4264:
4261:
4256:
4251:
4247:
4237:
4234:
4232:
4229:
4228:
4225:
4220:
4215:
4211:
4201:
4198:
4196:
4193:
4192:
4189:
4185:
4179:
4175:
4165:
4162:
4160:
4157:
4155:
4152:
4151:
4148:
4144:
4139:
4135:
4121:
4118:
4117:
4115:
4111:
4105:
4102:
4100:
4097:
4095:
4094:Polydivisible
4092:
4090:
4087:
4085:
4082:
4080:
4077:
4075:
4072:
4071:
4069:
4065:
4059:
4056:
4054:
4051:
4049:
4046:
4044:
4041:
4039:
4036:
4035:
4033:
4030:
4025:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3985:
3983:
3980:
3976:
3968:
3965:
3964:
3963:
3960:
3959:
3957:
3954:
3950:
3938:
3935:
3934:
3933:
3930:
3928:
3925:
3923:
3920:
3918:
3915:
3913:
3910:
3908:
3905:
3903:
3900:
3898:
3895:
3893:
3890:
3889:
3887:
3883:
3877:
3874:
3873:
3871:
3867:
3861:
3858:
3856:
3853:
3852:
3850:
3848:Digit product
3846:
3840:
3837:
3835:
3832:
3830:
3827:
3825:
3822:
3821:
3819:
3817:
3813:
3805:
3802:
3800:
3797:
3796:
3795:
3792:
3791:
3789:
3787:
3782:
3778:
3774:
3769:
3764:
3760:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3715:
3712:
3710:
3707:
3705:
3702:
3700:
3697:
3695:
3692:
3690:
3687:
3685:
3684:ErdĆsâNicolas
3682:
3680:
3677:
3675:
3672:
3671:
3668:
3663:
3659:
3653:
3649:
3635:
3632:
3630:
3627:
3626:
3624:
3622:
3618:
3612:
3609:
3607:
3604:
3602:
3599:
3597:
3594:
3593:
3591:
3589:
3585:
3579:
3576:
3574:
3571:
3569:
3566:
3564:
3561:
3559:
3556:
3554:
3551:
3550:
3548:
3546:
3542:
3536:
3533:
3531:
3528:
3527:
3525:
3523:
3519:
3513:
3510:
3508:
3505:
3503:
3502:Superabundant
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3483:
3480:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3404:
3402:
3400:
3396:
3392:
3388:
3384:
3379:
3375:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3306:
3303:
3299:
3294:
3290:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3261:
3258:
3254:
3249:
3245:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3136:
3133:
3126:
3122:
3104:
3101:
3099:
3096:
3094:
3091:
3090:
3088:
3084:
3081:
3079:
3078:4-dimensional
3075:
3065:
3062:
3061:
3059:
3057:
3053:
3047:
3044:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3018:
3016:
3014:
3010:
3004:
3001:
2999:
2996:
2994:
2991:
2989:
2988:Centered cube
2986:
2984:
2981:
2980:
2978:
2976:
2972:
2969:
2967:
2966:3-dimensional
2963:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2904:
2902:
2900:
2896:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2846:
2844:
2842:
2838:
2835:
2833:
2832:2-dimensional
2829:
2825:
2821:
2816:
2812:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2776:Nonhypotenuse
2774:
2773:
2770:
2763:
2759:
2749:
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2725:
2722:
2715:
2711:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2662:
2659:
2654:
2649:
2645:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2615:
2612:
2611:
2608:
2601:
2597:
2587:
2584:
2582:
2579:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2553:
2550:
2545:
2539:
2535:
2525:
2522:
2520:
2517:
2515:
2514:Perfect power
2512:
2510:
2507:
2505:
2504:Seventh power
2502:
2500:
2497:
2495:
2492:
2490:
2487:
2485:
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2465:
2462:
2460:
2457:
2456:
2453:
2448:
2443:
2439:
2435:
2427:
2422:
2420:
2415:
2413:
2408:
2407:
2404:
2392:
2389:
2387:
2384:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2348:
2346:
2342:
2336:
2333:
2331:
2330:Polydivisible
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2307:
2305:
2302:
2298:
2292:
2289:
2287:
2284:
2281:
2277:
2274:
2272:
2269:
2268:
2266:
2263:
2259:
2253:
2250:
2248:
2245:
2243:
2240:
2238:
2235:
2233:
2232:Superabundant
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2214:
2212:
2208:
2202:
2201:ErdĆsâNicolas
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2177:
2174:
2172:
2169:
2167:
2164:
2162:
2159:
2157:
2154:
2152:
2149:
2148:
2146:
2142:
2136:
2133:
2131:
2128:
2126:
2123:
2121:
2118:
2116:
2113:
2111:
2110:Perfect power
2108:
2106:
2103:
2101:
2098:
2096:
2093:
2091:
2088:
2086:
2083:
2081:
2078:
2076:
2073:
2072:
2070:
2066:
2061:
2051:
2048:
2046:
2043:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2022:
2020:
2016:
2007:
2002:
2000:
1995:
1993:
1988:
1987:
1984:
1978:
1974:
1972:
1968:
1966:
1962:
1960:
1956:
1953:
1949:
1946:
1942:
1939:
1935:
1932:
1928:
1925:
1921:
1920:
1919:
1917:
1913:
1910:(OEIS) lists
1909:
1899:
1898:
1893:
1890:
1885:
1884:
1880:
1875:
1874:
1869:
1866:
1864:
1860:
1856:
1852:
1851:
1847:
1831:
1824:
1817:
1814:
1809:
1802:
1799:
1785:
1780:
1776:
1769:
1762:
1759:
1750:
1749:
1744:
1738:
1735:
1729:
1722:
1719:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1686:
1682:
1678:
1674:
1668:
1665:
1660:
1659:
1653:
1649:
1643:
1640:
1629:
1625:
1619:
1616:
1605:
1601:
1595:
1592:
1587:
1586:
1580:
1576:
1570:
1567:
1562:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1526:
1523:
1512:
1508:
1501:
1498:
1486:
1485:GeeksforGeeks
1482:
1476:
1473:
1467:
1463:
1460:
1458:
1455:
1453:
1450:
1448:
1445:
1443:
1440:
1439:
1435:
1433:
1414:
1410:
1395:
1384:
1381:
1374:
1370:
1366:
1362:
1358:
1354:
1350:
1347:Note the set
1345:
1340:
1333:
1304:
1297:
1293:
1289:
1285:
1282:
1278:
1271:
1267:
1265:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1213:
1207:
1203:
1199:
1197:
1192:
1187:
1183:
1179:
1175:
1171:
1167:
1162:
1148:
1145:
1142:
1139:
1134:
1130:
1109:
1106:
1103:
1100:
1095:
1091:
1083:
1063:
1060:
1057:
1052:
1048:
1040:
1039:
1038:
1036:
1018:
1014:
1006:
1002:
998:
994:
990:
986:
978:
976:
959:
953:
933:
911:
907:
903:
899:
878:
858:
836:
832:
809:
805:
784:
762:
758:
735:
731:
725:
721:
717:
714:
705:
703:
699:
695:
690:
688:
669:
663:
639:
633:
630:
627:
623:
618:
614:
611:
605:
599:
596:
593:
590:
584:
581:
578:
565:
564:
563:
561:
557:
553:
549:
545:
541:
536:
522:
515:
496:
490:
467:
461:
458:
455:
450:
447:
444:
436:
433:
430:
426:
419:
413:
407:
403:
398:
392:
389:
386:
373:
372:
371:
354:
351:
348:
334:
329:
327:
323:
304:
301:
298:
280:
278:
276:
272:
268:
264:
260:
259:cryptanalysis
255:
253:
249:
248:Limit (music)
245:
241:
237:
232:
230:
226:
222:
218:
214:
206:
204:
190:
182:
178:
145:
143:
139:
135:
131:
130:
124:
122:
118:
114:
106:
105:prime factors
102:
94:
91:
83:
81:
79:
75:
71:
67:
63:
59:
58:prime factors
55:
51:
47:
45:
40:
38:
33:
32:number theory
19:
4058:Transposable
3922:Narcissistic
3829:Digital root
3749:Super-Poulet
3733:
3709:JordanâPĂłlya
3658:prime factor
3563:Noncototient
3530:Almost prime
3512:Superperfect
3487:Refactorable
3482:Quasiperfect
3457:Hyperperfect
3298:Pseudoprimes
3269:WallâSunâSun
3204:Ordered Bell
3174:FussâCatalan
3086:non-centered
3036:Dodecahedral
3013:non-centered
2899:non-centered
2801:Wolstenholme
2546:× 2 ± 1
2543:
2542:Of the form
2509:Eighth power
2489:Fourth power
2391:Superperfect
2386:Refactorable
2181:Superperfect
2176:Hyperperfect
2161:Quasiperfect
2119:
2045:Prime factor
1915:
1911:
1905:
1895:
1872:
1868:A. Granville
1854:
1848:Bibliography
1834:. Retrieved
1829:
1816:
1807:
1801:
1788:. Retrieved
1774:
1761:
1747:
1737:
1728:Music Review
1727:
1721:
1683:(3): 79â86,
1680:
1676:
1673:Aaboe, Asger
1667:
1655:
1642:
1631:. Retrieved
1627:
1618:
1607:. Retrieved
1603:
1594:
1582:
1569:
1542:
1538:
1534:
1525:
1514:. Retrieved
1510:
1500:
1489:. Retrieved
1487:. 2018-02-12
1484:
1475:
1452:Round number
1447:Rough number
1373:homomorphism
1348:
1346:
1338:
1331:
1302:
1295:
1291:
1287:
1283:
1276:
1274:
1269:
1257:
1249:
1233:
1229:
1225:
1221:
1211:
1205:
1204:-smooth and
1201:
1200:
1188:â (sequence
1185:
1177:
1173:
1169:
1165:
1163:
1081:
1079:
1034:
1004:
1001:ultrafriable
1000:
996:
992:
988:
984:
982:
797:-smooth and
706:
701:
693:
691:
655:
559:
555:
551:
547:
543:
539:
537:
482:
332:
330:
325:
321:
284:
281:Distribution
256:
244:music theory
234:5-smooth or
233:
224:
220:
210:
207:Applications
146:
137:
133:
128:
125:
120:
116:
112:
100:
87:
70:cryptography
61:
49:
43:
42:
36:
35:
29:
4079:Extravagant
4074:Equidigital
4029:permutation
3988:Palindromic
3962:Automorphic
3860:Sum-product
3839:Sum-product
3794:Persistence
3689:ErdĆsâWoods
3611:Untouchable
3492:Semiperfect
3442:Hemiperfect
3103:Tesseractic
3041:Icosahedral
3021:Tetrahedral
2952:Dodecagonal
2653:Recursively
2524:Prime power
2499:Sixth power
2494:Fifth power
2474:Power of 10
2432:Classes of
2315:Extravagant
2310:Equidigital
2271:Untouchable
2191:Semiperfect
2171:Hemiperfect
2100:Square-free
1830:math.vt.edu
1357:factor base
993:powersmooth
554:: that is,
74:powers of 2
4340:Categories
4291:Graphemics
4164:Pernicious
4018:Undulating
3993:Pandigital
3967:Trimorphic
3568:Nontotient
3417:Arithmetic
3031:Octahedral
2932:Heptagonal
2922:Pentagonal
2907:Triangular
2748:SierpiĆski
2670:Jacobsthal
2469:Power of 3
2464:Power of 2
2351:Arithmetic
2344:Other sets
2303:-dependent
1808:IWSEC 2015
1633:2019-12-12
1609:2019-12-12
1604:w3resource
1516:2019-12-12
1491:2019-12-12
1275:Moreover,
1210:Pollard's
987:is called
698:almost all
95:is called
84:Definition
4048:Parasitic
3897:Factorion
3824:Digit sum
3816:Digit sum
3634:Fortunate
3621:Primorial
3535:Semiprime
3472:Practical
3437:Descartes
3432:Deficient
3422:Betrothed
3264:Wieferich
3093:Pentatope
3056:pyramidal
2947:Decagonal
2942:Nonagonal
2937:Octagonal
2927:Hexagonal
2786:Practical
2733:Congruent
2665:Fibonacci
2629:Loeschian
2381:Descartes
2356:Deficient
2291:Betrothed
2196:Practical
2085:Semiprime
2080:Composite
1897:MathWorld
1784:0810.2067
1713:164195082
1402:→
1396:θ
1382:ϕ
1146:≰
1107:≰
1058:≤
1053:ν
1037:satisfy:
1033:dividing
1019:ν
983:Further,
954:ρ
704:-smooth.
664:ρ
631:
600:ρ
597:⋅
573:Ψ
562:. Then,
491:π
459:
448:
434:≤
427:∏
408:π
399:∼
381:Ψ
343:Ψ
293:Ψ
4120:Friedman
4053:Primeval
3998:Repdigit
3955:-related
3902:Kaprekar
3876:Meertens
3799:Additive
3786:dynamics
3694:Friendly
3606:Sociable
3596:Amicable
3407:Abundant
3387:dynamics
3209:Schröder
3199:Narayana
3169:Eulerian
3159:Delannoy
3154:Dedekind
2975:centered
2841:centered
2728:Amenable
2685:Narayana
2675:Leonardo
2571:Mersenne
2519:Powerful
2459:Achilles
2366:Solitary
2361:Friendly
2286:Sociable
2276:Amicable
2264:-related
2217:Abundant
2115:Achilles
2105:Powerful
2018:Overview
1954:(etc...)
1745:(1981),
1436:See also
1363:and the
1260:-smooth
692:For any
512:denotes
187:-smooth
90:positive
46:-friable
4293:related
4257:related
4221:related
4219:Sorting
4104:Vampire
4089:Harshad
4031:related
4003:Repunit
3917:Lychrel
3892:Dudeney
3744:StĂžrmer
3739:Sphenic
3724:Regular
3662:divisor
3601:Perfect
3497:Sublime
3467:Perfect
3194:Motzkin
3149:Catalan
2690:Padovan
2624:Leyland
2619:Idoneal
2614:Hilbert
2586:Woodall
2371:Sublime
2325:Harshad
2151:Perfect
2135:Unusual
2125:Regular
2095:Sphenic
2030:Divisor
1977:A080683
1971:A080682
1965:A080681
1959:A080197
1952:A051038
1945:A002473
1938:A051037
1931:A003586
1924:A000079
1836:26 July
1790:26 July
1705:0191779
1697:1359089
1650:(ed.).
1577:(ed.).
1194:in the
1191:A003418
1164:Unlike
685:is the
93:integer
54:integer
39:-smooth
4159:Odious
4084:Frugal
4038:Cyclic
4027:Digit-
3734:Smooth
3719:Pronic
3679:Cyclic
3656:Other
3629:Euclid
3279:Wilson
3253:Primes
2912:Square
2781:Polite
2743:Riesel
2738:Knödel
2700:Perrin
2581:Thabit
2566:Fermat
2556:Cullen
2479:Square
2447:Powers
2320:Frugal
2280:Triple
2120:Smooth
2090:Pronic
1947:(2357)
1861:
1711:
1703:
1695:
1557:
1353:subset
1254:groups
1252:)âfor
1005:powers
750:where
656:where
483:where
191:it is
179:, but
101:smooth
56:whose
52:is an
50:number
4200:Prime
4195:Lucky
4184:sieve
4113:Other
4099:Smith
3979:Digit
3937:Happy
3912:Keith
3885:Other
3729:Rough
3699:Giuga
3164:Euler
3026:Cubic
2680:Lucas
2576:Proth
2335:Smith
2252:Weird
2130:Rough
2075:Prime
1940:(235)
1826:(PDF)
1779:arXiv
1771:(PDF)
1752:(PDF)
1709:S2CID
1693:JSTOR
1262:order
246:(see
177:prime
34:, an
4154:Evil
3834:Self
3784:and
3674:Blum
3385:and
3189:Lobb
3144:Cake
3139:Bell
2889:Star
2796:Ulam
2695:Pell
2484:Cube
2301:Base
1933:(23)
1906:The
1859:ISBN
1838:2017
1792:2017
1656:The
1583:The
1555:ISBN
1541:)".
1230:n. A
1216:and
1196:OEIS
1122:and
1082:e.g.
995:(or
285:Let
119:and
41:(or
4272:Ban
3660:or
3179:Lah
1926:(2)
1918:s:
1685:doi
1547:doi
1359:of
1281:set
1256:of
1218:ECM
777:is
707:If
628:log
542:as
456:log
445:log
271:VSH
231:.)
175:is
30:In
4342::
1894:.
1870:,
1828:.
1777:.
1773:.
1707:,
1701:MR
1699:,
1691:,
1681:19
1679:,
1654:.
1626:.
1602:.
1581:.
1553:.
1535:GF
1509:.
1483:.
1432:.
1344:.
1264:.
1232:s
1143:16
975:.
696:,
689:.
535:.
370::
277:.
254:.
203:.
167:â„
144:.
115:,
88:A
80:.
48:)
2544:a
2425:e
2418:t
2411:v
2282:)
2278:(
2005:e
1998:t
1991:v
1916:B
1912:B
1900:.
1840:.
1810:.
1796:f
1794:.
1781::
1756:.
1732:.
1716:.
1687::
1636:.
1612:.
1563:.
1549::
1539:q
1537:(
1519:.
1494:.
1419:Z
1415:n
1411:/
1406:Z
1399:]
1393:[
1389:Z
1385::
1349:A
1342:3
1339:A
1335:3
1332:A
1317:Z
1306:2
1303:A
1299:1
1296:A
1292:A
1288:m
1284:A
1277:m
1270:A
1258:n
1250:n
1248:(
1246:O
1234:n
1226:n
1222:n
1212:p
1206:n
1202:n
1186:n
1178:n
1174:n
1170:n
1166:n
1149:5
1140:=
1135:4
1131:2
1110:5
1104:9
1101:=
1096:2
1092:3
1064:.
1061:n
1049:p
1035:m
1015:p
999:-
997:n
991:-
989:n
985:m
963:)
960:u
957:(
934:x
912:u
908:/
904:1
900:x
879:n
859:B
837:1
833:n
810:2
806:n
785:B
763:1
759:n
736:2
732:n
726:1
722:n
718:=
715:n
702:k
694:k
673:)
670:u
667:(
640:)
634:y
624:x
619:(
615:O
612:+
609:)
606:u
603:(
594:x
591:=
588:)
585:y
582:,
579:x
576:(
560:y
556:x
552:y
548:x
544:u
540:u
523:B
500:)
497:B
494:(
468:.
462:p
451:x
437:B
431:p
420:!
417:)
414:B
411:(
404:1
396:)
393:B
390:,
387:x
384:(
358:)
355:B
352:,
349:x
346:(
333:B
326:x
322:y
308:)
305:y
302:,
299:x
296:(
225:B
221:n
201:B
197:p
193:p
185:B
173:B
169:p
165:B
161:B
157:p
153:B
149:B
121:c
117:b
113:a
109:B
99:-
97:B
62:n
44:n
37:n
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.