4633:
2541:
numbering systems) but with the numbers of symbols it has. For example, it is much tougher to figure out whether 147 (CXLVII) is a
Friedman number in Roman numerals than it is to make the same determination for 1001 (MI). With Roman numerals, one can at least derive quite a few Friedman expressions
260:
127, 343, 736, 1285, 2187, 2502, 2592, 2737, 3125, 3685, 3864, 3972, 4096, 6455, 11264, 11664, 12850, 13825, 14641, 15552, 15585, 15612, 15613, 15617, 15618, 15621, 15622, 15623, 15624, 15626, 15632, 15633, 15642, 15645, 15655, 15656, 15662, 15667, 15688, 16377, 16384, 16447, 16875, 17536, 18432,
94:
127, 347, 2503, 12101, 12107, 12109, 15629, 15641, 15661, 15667, 15679, 16381, 16447, 16759, 16879, 19739, 21943, 27653, 28547, 28559, 29527, 29531, 32771, 32783, 35933, 36457, 39313, 39343, 43691, 45361, 46619, 46633, 46643, 46649, 46663, 46691, 48751, 48757, 49277, 58921, 59051, 59053, 59263,
287:
127, 15667, 16447, 19739, 28559, 32771, 39343, 46633, 46663, 117619, 117643, 117763, 125003, 131071, 137791, 147419, 156253, 156257, 156259, 229373, 248839, 262139, 262147, 279967, 294829, 295247, 326617, 466553, 466561, 466567, 585643, 592763, 649529, 728993, 759359, 786433, 937577 (sequence
2533:
Some research into Roman numeral
Friedman numbers for which the expression uses some of the other operators has been done. The first such nice Roman numeral Friedman number discovered was 8, since VIII = (V - I) Ă II. Other such nontrivial examples have been found.
56:
25, 121, 125, 126, 127, 128, 153, 216, 289, 343, 347, 625, 688, 736, 1022, 1024, 1206, 1255, 1260, 1285, 1296, 1395, 1435, 1503, 1530, 1792, 1827, 2048, 2187, 2349, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, ... (sequence
310:
tends to infinity. This result extends to
Friedman numbers under any base of representation. He also proved that the same is true also for binary, ternary and quaternary nice Friedman numbers. The case of base-10 nice Friedman numbers is still open.
2373:
48:. Here, non-trivial means that at least one operation besides concatenation is used. Leading zeros cannot be used, since that would also result in trivial Friedman numbers, such as 024 = 20 + 4. For example, 347 is a Friedman number in the
255:
Friedman number is a
Friedman number where the digits in the expression can be arranged to be in the same order as in the number itself. For example, we can arrange 127 = 2 â 1 as 127 = â1 + 2. The first nice Friedman numbers are:
2226:
2117:
2008:
1899:
1753:
1389:
1644:
1156:
1474:
1553:
2529:
with more than one symbol are
Friedman numbers. The expression is created by simply inserting + signs into the numeral, and occasionally the â sign with slight rearrangement of the order of the symbols.
782:
937:
1039:
2542:
from any new expression one discovers. Since 8 is a nice nontrivial nice Roman numeral
Friedman number, it follows that any number ending in VIII is also such a Friedman number.
2735:
2263:
302:
Michael Brand proved that the density of
Friedman numbers among the naturals is 1, which is to say that the probability of a number chosen randomly and uniformly between 1 and
1789:
1297:
2514:
325:
There usually are fewer 2-digit
Friedman numbers than 3-digit and more in any given base, but the 2-digit ones are easier to find. If we represent a 2-digit number as
2256:
846:
659:
2425:
2511:
1225:
986:
2642:
2485:
2465:
2445:
2396:
1179:
960:
805:
448:
The first few known
Friedman numbers in other small bases are shown below, written in their respective bases. Numbers shown in bold are nice Friedman numbers.
317:
are a subset of
Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example 1260 = 21 Ă 60.
4662:
4657:
295:
268:
102:
64:
2124:
2015:
1906:
1797:
1651:
2728:
1305:
1560:
1047:
1396:
3535:
2721:
2537:
The difficulty of finding nontrivial Friedman numbers in Roman numerals increases not with the size of the number (as is the case with
3530:
3545:
3525:
1481:
4238:
3818:
3540:
4324:
667:
3640:
3990:
3309:
3102:
1231:, 250068 = 500 + 68, from which we can easily deduce the range of consecutive Friedman numbers from 250000 to 250099 in
4025:
3995:
3670:
3660:
854:
4166:
3580:
3314:
3294:
3856:
4020:
4115:
3738:
3495:
3304:
3286:
3180:
3170:
3160:
4000:
4243:
3788:
3409:
3195:
3190:
3185:
3175:
3152:
3228:
2638:
3485:
4354:
4319:
4105:
4015:
3889:
3864:
3773:
3763:
3375:
3357:
3277:
994:
4614:
3884:
3758:
3389:
3165:
2945:
2872:
2368:{\displaystyle n\times 10^{50}+{\text{15AA51}}=n\times 10^{50}+(10+{\text{A}}/{\text{A}})^{5}+0+0+\ldots }
49:
3869:
3723:
3650:
2805:
1201:
0's, we can find sequences of consecutive Friedman numbers which are arbitrarily long. For example, for
4578:
4218:
2663:
4511:
4405:
4369:
4110:
3833:
3813:
3630:
3299:
3087:
3059:
1762:
4233:
4097:
4092:
4060:
3823:
3798:
3793:
3768:
3698:
3694:
3625:
3515:
3347:
3143:
3112:
2538:
1270:
4632:
4636:
4390:
4385:
4299:
4273:
4171:
4150:
3922:
3803:
3753:
3675:
3645:
3585:
3352:
3332:
3263:
2976:
76:
3520:
4530:
4475:
4329:
4304:
4278:
4055:
3733:
3728:
3655:
3635:
3620:
3342:
3324:
3243:
3233:
3218:
2996:
2981:
2235:
4566:
4359:
3945:
3917:
3907:
3899:
3783:
3748:
3743:
3710:
3404:
3367:
3258:
3253:
3248:
3238:
3210:
3097:
3049:
3044:
3001:
2940:
2675:
2590:
825:
629:
37:
72:
4542:
4364:
4290:
4213:
4187:
4005:
3718:
3575:
3510:
3480:
3470:
3465:
3131:
3039:
2986:
2830:
2770:
2401:
33:
2490:
1204:
965:
4547:
4415:
4400:
4264:
4228:
4203:
4079:
4050:
4035:
3912:
3808:
3778:
3505:
3460:
3337:
2935:
2930:
2925:
2897:
2882:
2795:
2780:
2758:
2745:
2613:
2470:
2450:
2430:
2381:
1164:
945:
790:
314:
41:
29:
1267:
There are an infinite number of prime Friedman numbers in all bases, because for base
4651:
4470:
4454:
4395:
4349:
4045:
4030:
3940:
3665:
3223:
3092:
3054:
3011:
2892:
2877:
2867:
2825:
2815:
2790:
2526:
442:
45:
4506:
4495:
4410:
4248:
4223:
4140:
4040:
4010:
3985:
3969:
3874:
3841:
3590:
3564:
3475:
3414:
2991:
2887:
2820:
2800:
2775:
87:
2650:
4465:
4340:
4145:
3609:
3500:
3455:
3450:
3200:
3107:
3006:
2835:
2810:
2785:
2221:{\displaystyle n\times 10^{60}+161051=n\times 10^{60}+(10+1-0)^{5}+0+0+\ldots }
2112:{\displaystyle n\times 10^{60}+162151=n\times 10^{60}+(10+2-1)^{5}+0+0+\ldots }
2003:{\displaystyle n\times 10^{60}+163251=n\times 10^{60}+(10+3-2)^{5}+0+0+\ldots }
1894:{\displaystyle n\times 10^{60}+164351=n\times 10^{60}+(10+4-3)^{5}+0+0+\ldots }
1748:{\displaystyle n\times 10^{13}+25352411=n\times 10^{2\times 5-1}+(5+2)^{(3+4)}}
4602:
4583:
3879:
3490:
2680:
2713:
36:
in combination with any of the four basic arithmetic operators (+, â, Ă, Ă·),
4208:
4135:
4127:
3932:
3846:
2964:
1384:{\displaystyle n\times 10^{1111}+11111111=n\times 10^{1111}+10^{1000}-1+0+0}
437:
Friedman numbers also exist for bases other than base 10. For example, 11001
4309:
1639:{\displaystyle n\times 10^{13}+2443111=n\times 10^{4+4}+(2\times 3)^{11}}
1260:
1238:
1151:{\displaystyle 2b+k=2\left({\frac {k(k-1)}{2}}\right)+k=k^{2}-k+k=k^{2}}
4314:
3973:
1253:
1232:
1228:
283:
Friedman number that's also prime. The first nice Friedman primes are:
21:
1256:
that is thought to be a Friedman number is 99999999 = (9 + 9/9) â 9/9.
1469:{\displaystyle n\times 10^{102}+1101221=n\times 10^{102}+2^{101}+0+0}
1246:
2697:
2576:
2688:
2630:
25:
4600:
4564:
4528:
4492:
4452:
4077:
3966:
3692:
3607:
3562:
3439:
3129:
3076:
3028:
2962:
2914:
2852:
2756:
2717:
1548:{\displaystyle n\times 10^{20}+310233=n\times 10^{20}+33^{3}+0}
381:
to see which ones are true. We need not concern ourselves with
32:, is the result of a non-trivial expression using all its own
2575:
Michael Brand, "On the Density of Nice Friedmans", Oct 2013,
2691:
2633:
290:
263:
97:
59:
2692:
sequence A119710 (Radical narcissistic numbers)
95:
59273, 64513, 74353, 74897, 78163, 83357, ... (sequence
114:
The expressions of the first few Friedman numbers are:
52:, since 347 = 7 + 4. The decimal Friedman numbers are:
2517:, the sequence contains an infinite number of primes.
2398:. The numbers of this form are an arithmetic sequence
1189:). From the observation that all numbers of the form 2
2698:"Pretty wild narcissistic numbers - numbers that pwn"
2493:
2473:
2453:
2433:
2404:
2384:
2266:
2238:
2127:
2018:
1909:
1800:
1765:
1654:
1563:
1484:
1399:
1308:
1273:
1207:
1167:
1050:
997:
968:
948:
857:
828:
793:
670:
632:
777:{\displaystyle b^{2}+mb+k=(mk-m+m)b+k=mbk+k=k(mb+1)}
349:â1, we need only check each possible combination of
4424:
4378:
4338:
4289:
4263:
4196:
4180:
4159:
4126:
4091:
3931:
3898:
3855:
3832:
3709:
3397:
3388:
3366:
3323:
3285:
3276:
3209:
3151:
3142:
2558:Michael Brand, "Friedman numbers have density 1",
2505:
2479:
2459:
2439:
2419:
2390:
2367:
2250:
2220:
2111:
2002:
1893:
1783:
1747:
1638:
1547:
1468:
1383:
1291:
1263:with at least 22 digits are nice Friedman numbers.
1219:
1173:
1150:
1033:
980:
954:
931:
840:
799:
776:
653:
932:{\displaystyle {(b^{n}+1)}^{2}=b^{2n}+2{b^{n}}+1}
2513:are always relatively prime, and therefore, by
2515:Dirichlet's theorem on arithmetic progressions
2729:
2643:The On-Line Encyclopedia of Integer Sequences
8:
2634:sequence A036057 (Friedman number)
2467:are relatively prime regardless of base as
4597:
4561:
4525:
4489:
4449:
4123:
4088:
4074:
3963:
3706:
3689:
3604:
3559:
3436:
3394:
3282:
3148:
3139:
3126:
3073:
3030:Possessing a specific set of other numbers
3025:
2959:
2911:
2849:
2753:
2736:
2722:
2714:
397:, since these will always be smaller than
2679:
2492:
2472:
2452:
2432:
2403:
2383:
2341:
2332:
2327:
2322:
2304:
2286:
2277:
2265:
2237:
2194:
2163:
2138:
2126:
2085:
2054:
2029:
2017:
1976:
1945:
1920:
1908:
1867:
1836:
1811:
1799:
1764:
1727:
1690:
1665:
1653:
1630:
1599:
1574:
1562:
1533:
1520:
1495:
1483:
1448:
1435:
1410:
1398:
1357:
1344:
1319:
1307:
1272:
1206:
1166:
1142:
1117:
1073:
1049:
1004:
996:
967:
947:
916:
911:
896:
883:
867:
859:
856:
827:
792:
675:
669:
631:
79:and recreational mathematics enthusiast.
75:, a now-retired mathematics professor at
450:
116:
2608:
2606:
2604:
2602:
2600:
2551:
1161:is a Friedman number (written in base
962:as 100...00200...001 = 100..001, with
942:is a Friedman number (written in base
787:is a Friedman number (written in base
306:to be a Friedman number tends to 1 as
1034:{\displaystyle b={\frac {k(k-1)}{2}}}
988:zeroes between each nonzero number).
7:
1249:that is a Friedman number is 33 = 3.
90:. The decimal Friedman primes are:
2566:(16â17), Nov. 2013, pp. 2389-2395.
86:is a Friedman number that is also
14:
2664:"Friedman numbers have density 1"
2657:. Problem of the Month. Aug 2000.
2591:"Friedman numbers in other bases"
441:= 25 is a Friedman number in the
71:Friedman numbers are named after
4663:Eponymous numbers in mathematics
4658:Base-dependent integer sequences
4631:
4239:Perfect digit-to-digit invariant
481:121, 221, 1022, 1122, 1211, ...
321:Finding 2-digit Friedman numbers
2577:https://arxiv.org/abs/1310.2390
579:121, 237, 24A, 1245, 1246, ...
563:121, 2A9, 603, 1163, 1533, ...
2702:Theoretical Research Institute
2338:
2313:
2191:
2172:
2082:
2063:
1973:
1954:
1864:
1845:
1784:{\displaystyle 7\leq b\leq 10}
1740:
1728:
1724:
1711:
1627:
1614:
1091:
1079:
1022:
1010:
879:
860:
771:
756:
720:
699:
261:19453, 19683, 19739 (sequence
1:
3078:Expressible via specific sums
2706:Extension to Friedman numbers
2378:are Friedman numbers for all
1292:{\displaystyle 2\leq b\leq 6}
615:121, 129, 145, 183, 27D, ...
571:121, 127, 135, 144, 163, ...
413:. The same clearly holds for
2668:Discrete Applied Mathematics
2560:Discrete Applied Mathematics
4167:Multiplicative digital root
2662:Brand, Michael (Nov 2013).
4679:
543:, 121, 125, 143, 251, ...
4627:
4610:
4596:
4574:
4560:
4538:
4524:
4502:
4488:
4461:
4448:
4244:Perfect digital invariant
4087:
4073:
3981:
3962:
3819:Superior highly composite
3705:
3688:
3616:
3603:
3571:
3558:
3446:
3435:
3138:
3125:
3083:
3072:
3035:
3024:
2972:
2958:
2921:
2910:
2863:
2848:
2766:
2752:
2681:10.1016/j.dam.2013.05.027
1252:The smallest repdigit in
1245:The smallest repdigit in
520:, 52, 121, 124, 133, ...
3857:Euler's totient function
3641:EulerâJacobi pseudoprime
2916:Other polynomial numbers
2525:In a trivial sense, all
1259:It has been proven that
493:, 1203, 1230, 1321, ...
473:, 1001111, 1010001, ...
3671:SomerâLucas pseudoprime
3661:LucasâCarmichael number
3496:Lazy caterer's sequence
2251:{\displaystyle b>10}
357:against the equalities
345:are integers from 0 to
3546:WedderburnâEtherington
2946:Lucky numbers of Euler
2639:Other Friedman numbers
2507:
2481:
2461:
2441:
2421:
2392:
2369:
2252:
2222:
2113:
2004:
1895:
1785:
1749:
1640:
1549:
1470:
1385:
1293:
1221:
1193:Ă b can be written as
1175:
1152:
1035:
982:
956:
933:
842:
841:{\displaystyle b>2}
801:
778:
655:
654:{\displaystyle b=mk-m}
50:decimal numeral system
3834:Prime omega functions
3651:Frobenius pseudoprime
3441:Combinatorial numbers
3310:Centered dodecahedral
3103:Primary pseudoperfect
2508:
2482:
2462:
2442:
2422:
2393:
2370:
2253:
2223:
2114:
2005:
1896:
1786:
1750:
1641:
1550:
1471:
1386:
1294:
1222:
1176:
1153:
1036:
983:
957:
934:
843:
802:
779:
656:
445:, since 11001 = 101.
443:binary numeral system
4293:-composition related
4093:Arithmetic functions
3695:Arithmetic functions
3631:Elliptic pseudoprime
3315:Centered icosahedral
3295:Centered tetrahedral
2674:(16â17): 2389â2395.
2521:Using Roman numerals
2491:
2471:
2451:
2431:
2420:{\displaystyle pn+q}
2402:
2382:
2264:
2236:
2125:
2016:
1907:
1798:
1763:
1652:
1561:
1482:
1397:
1306:
1271:
1205:
1165:
1048:
995:
966:
946:
855:
826:
791:
668:
630:
279:Friedman prime is a
4219:Kaprekar's constant
3739:Colossally abundant
3626:Catalan pseudoprime
3526:SchröderâHipparchus
3305:Centered octahedral
3181:Centered heptagonal
3171:Centered pentagonal
3161:Centered triangular
2761:and related numbers
2539:positional notation
2506:{\displaystyle b+1}
1220:{\displaystyle k=5}
981:{\displaystyle n-1}
603:26, 121, 136, 154,
4637:Mathematics portal
4579:Aronson's sequence
4325:SmarandacheâWellin
4082:-dependent numbers
3789:Primitive abundant
3676:Strong pseudoprime
3666:Perrin pseudoprime
3646:Fermat pseudoprime
3586:Wolstenholme prime
3410:Squared triangular
3196:Centered decagonal
3191:Centered nonagonal
3186:Centered octagonal
3176:Centered hexagonal
2651:"Friedman numbers"
2503:
2477:
2457:
2437:
2417:
2388:
2365:
2248:
2218:
2109:
2000:
1891:
1781:
1745:
1636:
1545:
1466:
1381:
1289:
1241:Friedman numbers:
1217:
1171:
1148:
1031:
978:
952:
929:
838:
797:
774:
651:
110:Results in base 10
77:Stetson University
4645:
4644:
4623:
4622:
4592:
4591:
4556:
4555:
4520:
4519:
4484:
4483:
4444:
4443:
4440:
4439:
4259:
4258:
4069:
4068:
3958:
3957:
3954:
3953:
3900:Aliquot sequences
3711:Divisor functions
3684:
3683:
3656:Lucas pseudoprime
3636:Euler pseudoprime
3621:Carmichael number
3599:
3598:
3554:
3553:
3431:
3430:
3427:
3426:
3423:
3422:
3384:
3383:
3272:
3271:
3229:Square triangular
3121:
3120:
3068:
3067:
3020:
3019:
2954:
2953:
2906:
2905:
2844:
2843:
2589:Friedman, Erich.
2480:{\displaystyle b}
2460:{\displaystyle q}
2440:{\displaystyle p}
2391:{\displaystyle n}
2335:
2325:
2289:
1174:{\displaystyle b}
1098:
1029:
955:{\displaystyle b}
800:{\displaystyle b}
619:
618:
555:, 628, 1304, ...
457:Friedman numbers
249:
248:
38:additive inverses
4670:
4635:
4598:
4567:Natural language
4562:
4526:
4494:Generated via a
4490:
4450:
4355:Digit-reassembly
4320:Self-descriptive
4124:
4089:
4075:
4026:LucasâCarmichael
4016:Harmonic divisor
3964:
3890:Sparsely totient
3865:Highly cototient
3774:Multiply perfect
3764:Highly composite
3707:
3690:
3605:
3560:
3541:Telephone number
3437:
3395:
3376:Square pyramidal
3358:Stella octangula
3283:
3149:
3140:
3132:Figurate numbers
3127:
3074:
3026:
2960:
2912:
2850:
2754:
2738:
2731:
2724:
2715:
2708:
2690:
2685:
2683:
2658:
2632:
2618:
2617:
2610:
2595:
2594:
2586:
2580:
2573:
2567:
2556:
2512:
2510:
2509:
2504:
2486:
2484:
2483:
2478:
2466:
2464:
2463:
2458:
2446:
2444:
2443:
2438:
2426:
2424:
2423:
2418:
2397:
2395:
2394:
2389:
2374:
2372:
2371:
2366:
2346:
2345:
2336:
2333:
2331:
2326:
2323:
2309:
2308:
2290:
2287:
2282:
2281:
2257:
2255:
2254:
2249:
2227:
2225:
2224:
2219:
2199:
2198:
2168:
2167:
2143:
2142:
2118:
2116:
2115:
2110:
2090:
2089:
2059:
2058:
2034:
2033:
2009:
2007:
2006:
2001:
1981:
1980:
1950:
1949:
1925:
1924:
1900:
1898:
1897:
1892:
1872:
1871:
1841:
1840:
1816:
1815:
1790:
1788:
1787:
1782:
1754:
1752:
1751:
1746:
1744:
1743:
1707:
1706:
1670:
1669:
1645:
1643:
1642:
1637:
1635:
1634:
1610:
1609:
1579:
1578:
1554:
1552:
1551:
1546:
1538:
1537:
1525:
1524:
1500:
1499:
1475:
1473:
1472:
1467:
1453:
1452:
1440:
1439:
1415:
1414:
1390:
1388:
1387:
1382:
1362:
1361:
1349:
1348:
1324:
1323:
1298:
1296:
1295:
1290:
1226:
1224:
1223:
1218:
1180:
1178:
1177:
1172:
1157:
1155:
1154:
1149:
1147:
1146:
1122:
1121:
1103:
1099:
1094:
1074:
1040:
1038:
1037:
1032:
1030:
1025:
1005:
987:
985:
984:
979:
961:
959:
958:
953:
938:
936:
935:
930:
922:
921:
920:
904:
903:
888:
887:
882:
872:
871:
847:
845:
844:
839:
806:
804:
803:
798:
783:
781:
780:
775:
680:
679:
660:
658:
657:
652:
532:, 264, 514, ...
451:
337:is the base and
293:
266:
117:
100:
62:
4678:
4677:
4673:
4672:
4671:
4669:
4668:
4667:
4648:
4647:
4646:
4641:
4619:
4615:Strobogrammatic
4606:
4588:
4570:
4552:
4534:
4516:
4498:
4480:
4457:
4436:
4420:
4379:Divisor-related
4374:
4334:
4285:
4255:
4192:
4176:
4155:
4122:
4095:
4083:
4065:
3977:
3976:related numbers
3950:
3927:
3894:
3885:Perfect totient
3851:
3828:
3759:Highly abundant
3701:
3680:
3612:
3595:
3567:
3550:
3536:Stirling second
3442:
3419:
3380:
3362:
3319:
3268:
3205:
3166:Centered square
3134:
3117:
3079:
3064:
3031:
3016:
2968:
2967:defined numbers
2950:
2917:
2902:
2873:Double Mersenne
2859:
2840:
2762:
2748:
2746:natural numbers
2742:
2696:
2661:
2649:
2627:
2622:
2621:
2612:
2611:
2598:
2588:
2587:
2583:
2574:
2570:
2557:
2553:
2548:
2523:
2489:
2488:
2469:
2468:
2449:
2448:
2429:
2428:
2400:
2399:
2380:
2379:
2337:
2300:
2273:
2262:
2261:
2234:
2233:
2190:
2159:
2134:
2123:
2122:
2081:
2050:
2025:
2014:
2013:
1972:
1941:
1916:
1905:
1904:
1863:
1832:
1807:
1796:
1795:
1761:
1760:
1723:
1686:
1661:
1650:
1649:
1626:
1595:
1570:
1559:
1558:
1529:
1516:
1491:
1480:
1479:
1444:
1431:
1406:
1395:
1394:
1353:
1340:
1315:
1304:
1303:
1269:
1268:
1203:
1202:
1197:000...000 with
1163:
1162:
1138:
1113:
1075:
1069:
1046:
1045:
1006:
993:
992:
964:
963:
944:
943:
912:
892:
863:
858:
853:
852:
824:
823:
789:
788:
671:
666:
665:
628:
627:
624:
622:General results
440:
435:
323:
315:Vampire numbers
289:
262:
112:
96:
58:
40:, parentheses,
18:Friedman number
12:
11:
5:
4676:
4674:
4666:
4665:
4660:
4650:
4649:
4643:
4642:
4640:
4639:
4628:
4625:
4624:
4621:
4620:
4618:
4617:
4611:
4608:
4607:
4601:
4594:
4593:
4590:
4589:
4587:
4586:
4581:
4575:
4572:
4571:
4565:
4558:
4557:
4554:
4553:
4551:
4550:
4548:Sorting number
4545:
4543:Pancake number
4539:
4536:
4535:
4529:
4522:
4521:
4518:
4517:
4515:
4514:
4509:
4503:
4500:
4499:
4493:
4486:
4485:
4482:
4481:
4479:
4478:
4473:
4468:
4462:
4459:
4458:
4455:Binary numbers
4453:
4446:
4445:
4442:
4441:
4438:
4437:
4435:
4434:
4428:
4426:
4422:
4421:
4419:
4418:
4413:
4408:
4403:
4398:
4393:
4388:
4382:
4380:
4376:
4375:
4373:
4372:
4367:
4362:
4357:
4352:
4346:
4344:
4336:
4335:
4333:
4332:
4327:
4322:
4317:
4312:
4307:
4302:
4296:
4294:
4287:
4286:
4284:
4283:
4282:
4281:
4270:
4268:
4265:P-adic numbers
4261:
4260:
4257:
4256:
4254:
4253:
4252:
4251:
4241:
4236:
4231:
4226:
4221:
4216:
4211:
4206:
4200:
4198:
4194:
4193:
4191:
4190:
4184:
4182:
4181:Coding-related
4178:
4177:
4175:
4174:
4169:
4163:
4161:
4157:
4156:
4154:
4153:
4148:
4143:
4138:
4132:
4130:
4121:
4120:
4119:
4118:
4116:Multiplicative
4113:
4102:
4100:
4085:
4084:
4080:Numeral system
4078:
4071:
4070:
4067:
4066:
4064:
4063:
4058:
4053:
4048:
4043:
4038:
4033:
4028:
4023:
4018:
4013:
4008:
4003:
3998:
3993:
3988:
3982:
3979:
3978:
3967:
3960:
3959:
3956:
3955:
3952:
3951:
3949:
3948:
3943:
3937:
3935:
3929:
3928:
3926:
3925:
3920:
3915:
3910:
3904:
3902:
3896:
3895:
3893:
3892:
3887:
3882:
3877:
3872:
3870:Highly totient
3867:
3861:
3859:
3853:
3852:
3850:
3849:
3844:
3838:
3836:
3830:
3829:
3827:
3826:
3821:
3816:
3811:
3806:
3801:
3796:
3791:
3786:
3781:
3776:
3771:
3766:
3761:
3756:
3751:
3746:
3741:
3736:
3731:
3726:
3724:Almost perfect
3721:
3715:
3713:
3703:
3702:
3693:
3686:
3685:
3682:
3681:
3679:
3678:
3673:
3668:
3663:
3658:
3653:
3648:
3643:
3638:
3633:
3628:
3623:
3617:
3614:
3613:
3608:
3601:
3600:
3597:
3596:
3594:
3593:
3588:
3583:
3578:
3572:
3569:
3568:
3563:
3556:
3555:
3552:
3551:
3549:
3548:
3543:
3538:
3533:
3531:Stirling first
3528:
3523:
3518:
3513:
3508:
3503:
3498:
3493:
3488:
3483:
3478:
3473:
3468:
3463:
3458:
3453:
3447:
3444:
3443:
3440:
3433:
3432:
3429:
3428:
3425:
3424:
3421:
3420:
3418:
3417:
3412:
3407:
3401:
3399:
3392:
3386:
3385:
3382:
3381:
3379:
3378:
3372:
3370:
3364:
3363:
3361:
3360:
3355:
3350:
3345:
3340:
3335:
3329:
3327:
3321:
3320:
3318:
3317:
3312:
3307:
3302:
3297:
3291:
3289:
3280:
3274:
3273:
3270:
3269:
3267:
3266:
3261:
3256:
3251:
3246:
3241:
3236:
3231:
3226:
3221:
3215:
3213:
3207:
3206:
3204:
3203:
3198:
3193:
3188:
3183:
3178:
3173:
3168:
3163:
3157:
3155:
3146:
3136:
3135:
3130:
3123:
3122:
3119:
3118:
3116:
3115:
3110:
3105:
3100:
3095:
3090:
3084:
3081:
3080:
3077:
3070:
3069:
3066:
3065:
3063:
3062:
3057:
3052:
3047:
3042:
3036:
3033:
3032:
3029:
3022:
3021:
3018:
3017:
3015:
3014:
3009:
3004:
2999:
2994:
2989:
2984:
2979:
2973:
2970:
2969:
2963:
2956:
2955:
2952:
2951:
2949:
2948:
2943:
2938:
2933:
2928:
2922:
2919:
2918:
2915:
2908:
2907:
2904:
2903:
2901:
2900:
2895:
2890:
2885:
2880:
2875:
2870:
2864:
2861:
2860:
2853:
2846:
2845:
2842:
2841:
2839:
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2788:
2783:
2778:
2773:
2767:
2764:
2763:
2757:
2750:
2749:
2743:
2741:
2740:
2733:
2726:
2718:
2712:
2711:
2710:
2709:
2686:
2659:
2647:
2646:
2645:
2626:
2625:External links
2623:
2620:
2619:
2596:
2581:
2568:
2550:
2549:
2547:
2544:
2527:Roman numerals
2522:
2519:
2502:
2499:
2496:
2476:
2456:
2436:
2416:
2413:
2410:
2407:
2387:
2376:
2375:
2364:
2361:
2358:
2355:
2352:
2349:
2344:
2340:
2330:
2321:
2318:
2315:
2312:
2307:
2303:
2299:
2296:
2293:
2285:
2280:
2276:
2272:
2269:
2247:
2244:
2241:
2230:
2229:
2217:
2214:
2211:
2208:
2205:
2202:
2197:
2193:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2166:
2162:
2158:
2155:
2152:
2149:
2146:
2141:
2137:
2133:
2130:
2120:
2108:
2105:
2102:
2099:
2096:
2093:
2088:
2084:
2080:
2077:
2074:
2071:
2068:
2065:
2062:
2057:
2053:
2049:
2046:
2043:
2040:
2037:
2032:
2028:
2024:
2021:
2011:
1999:
1996:
1993:
1990:
1987:
1984:
1979:
1975:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1948:
1944:
1940:
1937:
1934:
1931:
1928:
1923:
1919:
1915:
1912:
1902:
1890:
1887:
1884:
1881:
1878:
1875:
1870:
1866:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1839:
1835:
1831:
1828:
1825:
1822:
1819:
1814:
1810:
1806:
1803:
1780:
1777:
1774:
1771:
1768:
1757:
1756:
1742:
1739:
1736:
1733:
1730:
1726:
1722:
1719:
1716:
1713:
1710:
1705:
1702:
1699:
1696:
1693:
1689:
1685:
1682:
1679:
1676:
1673:
1668:
1664:
1660:
1657:
1647:
1633:
1629:
1625:
1622:
1619:
1616:
1613:
1608:
1605:
1602:
1598:
1594:
1591:
1588:
1585:
1582:
1577:
1573:
1569:
1566:
1556:
1544:
1541:
1536:
1532:
1528:
1523:
1519:
1515:
1512:
1509:
1506:
1503:
1498:
1494:
1490:
1487:
1477:
1465:
1462:
1459:
1456:
1451:
1447:
1443:
1438:
1434:
1430:
1427:
1424:
1421:
1418:
1413:
1409:
1405:
1402:
1392:
1380:
1377:
1374:
1371:
1368:
1365:
1360:
1356:
1352:
1347:
1343:
1339:
1336:
1333:
1330:
1327:
1322:
1318:
1314:
1311:
1288:
1285:
1282:
1279:
1276:
1265:
1264:
1257:
1250:
1216:
1213:
1210:
1170:
1159:
1158:
1145:
1141:
1137:
1134:
1131:
1128:
1125:
1120:
1116:
1112:
1109:
1106:
1102:
1097:
1093:
1090:
1087:
1084:
1081:
1078:
1072:
1068:
1065:
1062:
1059:
1056:
1053:
1028:
1024:
1021:
1018:
1015:
1012:
1009:
1003:
1000:
977:
974:
971:
951:
940:
939:
928:
925:
919:
915:
910:
907:
902:
899:
895:
891:
886:
881:
878:
875:
870:
866:
862:
837:
834:
831:
796:
785:
784:
773:
770:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
678:
674:
650:
647:
644:
641:
638:
635:
623:
620:
617:
616:
613:
609:
608:
601:
597:
596:
585:
581:
580:
577:
573:
572:
569:
565:
564:
561:
557:
556:
549:
545:
544:
538:
534:
533:
526:
522:
521:
515:
511:
510:
505:, 1232, 1241,
499:
495:
494:
487:
483:
482:
479:
475:
474:
463:
459:
458:
455:
438:
434:
431:
322:
319:
300:
299:
273:
272:
247:
246:
243:
240:
237:
234:
231:
228:
225:
221:
220:
217:
214:
211:
208:
205:
202:
199:
195:
194:
191:
188:
185:
182:
179:
176:
173:
169:
168:
165:
162:
159:
156:
153:
150:
147:
143:
142:
139:
136:
133:
130:
127:
124:
121:
111:
108:
107:
106:
84:Friedman prime
73:Erich Friedman
69:
68:
42:exponentiation
30:numeral system
13:
10:
9:
6:
4:
3:
2:
4675:
4664:
4661:
4659:
4656:
4655:
4653:
4638:
4634:
4630:
4629:
4626:
4616:
4613:
4612:
4609:
4604:
4599:
4595:
4585:
4582:
4580:
4577:
4576:
4573:
4568:
4563:
4559:
4549:
4546:
4544:
4541:
4540:
4537:
4532:
4527:
4523:
4513:
4510:
4508:
4505:
4504:
4501:
4497:
4491:
4487:
4477:
4474:
4472:
4469:
4467:
4464:
4463:
4460:
4456:
4451:
4447:
4433:
4430:
4429:
4427:
4423:
4417:
4414:
4412:
4409:
4407:
4406:Polydivisible
4404:
4402:
4399:
4397:
4394:
4392:
4389:
4387:
4384:
4383:
4381:
4377:
4371:
4368:
4366:
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4347:
4345:
4342:
4337:
4331:
4328:
4326:
4323:
4321:
4318:
4316:
4313:
4311:
4308:
4306:
4303:
4301:
4298:
4297:
4295:
4292:
4288:
4280:
4277:
4276:
4275:
4272:
4271:
4269:
4266:
4262:
4250:
4247:
4246:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4220:
4217:
4215:
4212:
4210:
4207:
4205:
4202:
4201:
4199:
4195:
4189:
4186:
4185:
4183:
4179:
4173:
4170:
4168:
4165:
4164:
4162:
4160:Digit product
4158:
4152:
4149:
4147:
4144:
4142:
4139:
4137:
4134:
4133:
4131:
4129:
4125:
4117:
4114:
4112:
4109:
4108:
4107:
4104:
4103:
4101:
4099:
4094:
4090:
4086:
4081:
4076:
4072:
4062:
4059:
4057:
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4032:
4029:
4027:
4024:
4022:
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3997:
3996:ErdĆsâNicolas
3994:
3992:
3989:
3987:
3984:
3983:
3980:
3975:
3971:
3965:
3961:
3947:
3944:
3942:
3939:
3938:
3936:
3934:
3930:
3924:
3921:
3919:
3916:
3914:
3911:
3909:
3906:
3905:
3903:
3901:
3897:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3871:
3868:
3866:
3863:
3862:
3860:
3858:
3854:
3848:
3845:
3843:
3840:
3839:
3837:
3835:
3831:
3825:
3822:
3820:
3817:
3815:
3814:Superabundant
3812:
3810:
3807:
3805:
3802:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3730:
3727:
3725:
3722:
3720:
3717:
3716:
3714:
3712:
3708:
3704:
3700:
3696:
3691:
3687:
3677:
3674:
3672:
3669:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3629:
3627:
3624:
3622:
3619:
3618:
3615:
3611:
3606:
3602:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3573:
3570:
3566:
3561:
3557:
3547:
3544:
3542:
3539:
3537:
3534:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3459:
3457:
3454:
3452:
3449:
3448:
3445:
3438:
3434:
3416:
3413:
3411:
3408:
3406:
3403:
3402:
3400:
3396:
3393:
3391:
3390:4-dimensional
3387:
3377:
3374:
3373:
3371:
3369:
3365:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3330:
3328:
3326:
3322:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3300:Centered cube
3298:
3296:
3293:
3292:
3290:
3288:
3284:
3281:
3279:
3278:3-dimensional
3275:
3265:
3262:
3260:
3257:
3255:
3252:
3250:
3247:
3245:
3242:
3240:
3237:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3216:
3214:
3212:
3208:
3202:
3199:
3197:
3194:
3192:
3189:
3187:
3184:
3182:
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3162:
3159:
3158:
3156:
3154:
3150:
3147:
3145:
3144:2-dimensional
3141:
3137:
3133:
3128:
3124:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3088:Nonhypotenuse
3086:
3085:
3082:
3075:
3071:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3037:
3034:
3027:
3023:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2975:
2974:
2971:
2966:
2961:
2957:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2927:
2924:
2923:
2920:
2913:
2909:
2899:
2896:
2894:
2891:
2889:
2886:
2884:
2881:
2879:
2876:
2874:
2871:
2869:
2866:
2865:
2862:
2857:
2851:
2847:
2837:
2834:
2832:
2829:
2827:
2826:Perfect power
2824:
2822:
2819:
2817:
2816:Seventh power
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2768:
2765:
2760:
2755:
2751:
2747:
2739:
2734:
2732:
2727:
2725:
2720:
2719:
2716:
2707:
2703:
2699:
2695:
2694:
2693:
2687:
2682:
2677:
2673:
2669:
2665:
2660:
2656:
2652:
2648:
2644:
2640:
2637:
2636:
2635:
2629:
2628:
2624:
2615:
2609:
2607:
2605:
2603:
2601:
2597:
2592:
2585:
2582:
2578:
2572:
2569:
2565:
2561:
2555:
2552:
2545:
2543:
2540:
2535:
2531:
2528:
2520:
2518:
2516:
2500:
2497:
2494:
2474:
2454:
2434:
2414:
2411:
2408:
2405:
2385:
2362:
2359:
2356:
2353:
2350:
2347:
2342:
2328:
2319:
2316:
2310:
2305:
2301:
2297:
2294:
2291:
2283:
2278:
2274:
2270:
2267:
2260:
2259:
2258:
2245:
2242:
2239:
2232:and for base
2215:
2212:
2209:
2206:
2203:
2200:
2195:
2187:
2184:
2181:
2178:
2175:
2169:
2164:
2160:
2156:
2153:
2150:
2147:
2144:
2139:
2135:
2131:
2128:
2121:
2106:
2103:
2100:
2097:
2094:
2091:
2086:
2078:
2075:
2072:
2069:
2066:
2060:
2055:
2051:
2047:
2044:
2041:
2038:
2035:
2030:
2026:
2022:
2019:
2012:
1997:
1994:
1991:
1988:
1985:
1982:
1977:
1969:
1966:
1963:
1960:
1957:
1951:
1946:
1942:
1938:
1935:
1932:
1929:
1926:
1921:
1917:
1913:
1910:
1903:
1888:
1885:
1882:
1879:
1876:
1873:
1868:
1860:
1857:
1854:
1851:
1848:
1842:
1837:
1833:
1829:
1826:
1823:
1820:
1817:
1812:
1808:
1804:
1801:
1794:
1793:
1792:
1778:
1775:
1772:
1769:
1766:
1737:
1734:
1731:
1720:
1717:
1714:
1708:
1703:
1700:
1697:
1694:
1691:
1687:
1683:
1680:
1677:
1674:
1671:
1666:
1662:
1658:
1655:
1648:
1631:
1623:
1620:
1617:
1611:
1606:
1603:
1600:
1596:
1592:
1589:
1586:
1583:
1580:
1575:
1571:
1567:
1564:
1557:
1542:
1539:
1534:
1530:
1526:
1521:
1517:
1513:
1510:
1507:
1504:
1501:
1496:
1492:
1488:
1485:
1478:
1463:
1460:
1457:
1454:
1449:
1445:
1441:
1436:
1432:
1428:
1425:
1422:
1419:
1416:
1411:
1407:
1403:
1400:
1393:
1378:
1375:
1372:
1369:
1366:
1363:
1358:
1354:
1350:
1345:
1341:
1337:
1334:
1331:
1328:
1325:
1320:
1316:
1312:
1309:
1302:
1301:
1300:
1286:
1283:
1280:
1277:
1274:
1262:
1258:
1255:
1251:
1248:
1244:
1243:
1242:
1240:
1236:
1234:
1230:
1214:
1211:
1208:
1200:
1196:
1192:
1188:
1184:
1168:
1143:
1139:
1135:
1132:
1129:
1126:
1123:
1118:
1114:
1110:
1107:
1104:
1100:
1095:
1088:
1085:
1082:
1076:
1070:
1066:
1063:
1060:
1057:
1054:
1051:
1044:
1043:
1042:
1026:
1019:
1016:
1013:
1007:
1001:
998:
989:
975:
972:
969:
949:
926:
923:
917:
913:
908:
905:
900:
897:
893:
889:
884:
876:
873:
868:
864:
851:
850:
849:
835:
832:
829:
820:
818:
814:
810:
794:
768:
765:
762:
759:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
723:
717:
714:
711:
708:
705:
702:
696:
693:
690:
687:
684:
681:
676:
672:
664:
663:
662:
648:
645:
642:
639:
636:
633:
621:
614:
611:
610:
606:
602:
599:
598:
594:
590:
586:
583:
582:
578:
575:
574:
570:
567:
566:
562:
559:
558:
554:
550:
547:
546:
542:
539:
536:
535:
531:
527:
524:
523:
519:
516:
513:
512:
508:
504:
500:
497:
496:
492:
488:
485:
484:
480:
477:
476:
472:
468:
464:
461:
460:
456:
453:
452:
449:
446:
444:
432:
430:
428:
424:
420:
416:
412:
408:
404:
400:
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
340:
336:
332:
328:
320:
318:
316:
312:
309:
305:
297:
292:
286:
285:
284:
282:
278:
270:
265:
259:
258:
257:
254:
244:
241:
238:
235:
232:
229:
226:
223:
222:
218:
215:
212:
209:
206:
203:
200:
197:
196:
192:
189:
186:
183:
180:
177:
174:
171:
170:
166:
163:
160:
157:
154:
151:
148:
145:
144:
140:
137:
134:
131:
128:
125:
122:
119:
118:
115:
109:
104:
99:
93:
92:
91:
89:
85:
80:
78:
74:
66:
61:
55:
54:
53:
51:
47:
46:concatenation
43:
39:
35:
31:
27:
23:
19:
4431:
4370:Transposable
4234:Narcissistic
4141:Digital root
4061:Super-Poulet
4021:JordanâPĂłlya
3970:prime factor
3875:Noncototient
3842:Almost prime
3824:Superperfect
3799:Refactorable
3794:Quasiperfect
3769:Hyperperfect
3610:Pseudoprimes
3581:WallâSunâSun
3516:Ordered Bell
3486:FussâCatalan
3398:non-centered
3348:Dodecahedral
3325:non-centered
3211:non-centered
3113:Wolstenholme
2858:× 2 ± 1
2855:
2854:Of the form
2821:Eighth power
2801:Fourth power
2705:
2701:
2671:
2667:
2654:
2614:"Math Magic"
2584:
2571:
2563:
2559:
2554:
2536:
2532:
2524:
2377:
2231:
1791:the numbers
1758:
1299:the numbers
1266:
1237:
1198:
1194:
1190:
1186:
1182:
1160:
990:
941:
821:
816:
812:
808:
786:
625:
604:
592:
588:
552:
540:
529:
517:
506:
502:
490:
470:
466:
447:
436:
426:
422:
418:
414:
410:
406:
402:
398:
394:
390:
386:
382:
378:
374:
370:
366:
362:
358:
354:
350:
346:
342:
338:
334:
330:
326:
324:
313:
307:
303:
301:
280:
276:
274:
252:
250:
113:
83:
81:
70:
17:
15:
4391:Extravagant
4386:Equidigital
4341:permutation
4300:Palindromic
4274:Automorphic
4172:Sum-product
4151:Sum-product
4106:Persistence
4001:ErdĆsâWoods
3923:Untouchable
3804:Semiperfect
3754:Hemiperfect
3415:Tesseractic
3353:Icosahedral
3333:Tetrahedral
3264:Dodecagonal
2965:Recursively
2836:Prime power
2811:Sixth power
2806:Fifth power
2786:Power of 10
2744:Classes of
2228:in base 10,
595:, 173, ...
433:Other bases
141:expression
135:expression
129:expression
123:expression
28:in a given
26:represented
4652:Categories
4603:Graphemics
4476:Pernicious
4330:Undulating
4305:Pandigital
4279:Trimorphic
3880:Nontotient
3729:Arithmetic
3343:Octahedral
3244:Heptagonal
3234:Pentagonal
3219:Triangular
3060:SierpiĆski
2982:Jacobsthal
2781:Power of 3
2776:Power of 2
2546:References
2119:in base 9,
2010:in base 8,
1901:in base 7,
587:121, 128,
551:121, 134,
528:121, 143,
4360:Parasitic
4209:Factorion
4136:Digit sum
4128:Digit sum
3946:Fortunate
3933:Primorial
3847:Semiprime
3784:Practical
3749:Descartes
3744:Deficient
3734:Betrothed
3576:Wieferich
3405:Pentatope
3368:pyramidal
3259:Decagonal
3254:Nonagonal
3249:Octagonal
3239:Hexagonal
3098:Practical
3045:Congruent
2977:Fibonacci
2941:Loeschian
2363:…
2298:×
2271:×
2216:…
2185:−
2157:×
2132:×
2107:…
2076:−
2048:×
2023:×
1998:…
1967:−
1939:×
1914:×
1889:…
1858:−
1830:×
1805:×
1776:≤
1770:≤
1759:for base
1755:in base 6
1701:−
1695:×
1684:×
1659:×
1646:in base 5
1621:×
1593:×
1568:×
1555:in base 4
1514:×
1489:×
1476:in base 3
1429:×
1404:×
1391:in base 2
1364:−
1338:×
1313:×
1284:≤
1278:≤
1261:repdigits
1124:−
1086:−
1017:−
973:−
709:−
646:−
4432:Friedman
4365:Primeval
4310:Repdigit
4267:-related
4214:Kaprekar
4188:Meertens
4111:Additive
4098:dynamics
4006:Friendly
3918:Sociable
3908:Amicable
3719:Abundant
3699:dynamics
3521:Schröder
3511:Narayana
3481:Eulerian
3471:Delannoy
3466:Dedekind
3287:centered
3153:centered
3040:Amenable
2997:Narayana
2987:Leonardo
2883:Mersenne
2831:Powerful
2771:Achilles
2427:, where
1675:25352411
1329:11111111
1239:Repdigit
1227:, or in
991:In base
822:In base
626:In base
333:, where
24:, which
4605:related
4569:related
4533:related
4531:Sorting
4416:Vampire
4401:Harshad
4343:related
4315:Repunit
4229:Lychrel
4204:Dudeney
4056:StĂžrmer
4051:Sphenic
4036:Regular
3974:divisor
3913:Perfect
3809:Sublime
3779:Perfect
3506:Motzkin
3461:Catalan
3002:Padovan
2936:Leyland
2931:Idoneal
2926:Hilbert
2898:Woodall
1584:2443111
1420:1101221
1254:base 10
1233:base 10
1229:base 10
465:11001,
294:in the
291:A252483
267:in the
264:A080035
138:number
132:number
126:number
120:number
101:in the
98:A112419
63:in the
60:A036057
22:integer
4471:Odious
4396:Frugal
4350:Cyclic
4339:Digit-
4046:Smooth
4031:Pronic
3991:Cyclic
3968:Other
3941:Euclid
3591:Wilson
3565:Primes
3224:Square
3093:Polite
3055:Riesel
3050:Knödel
3012:Perrin
2893:Thabit
2878:Fermat
2868:Cullen
2791:Square
2759:Powers
2655:Github
2288:15AA51
2148:161051
2039:162151
1930:163251
1821:164351
1505:310233
1247:base 8
607:, ...
509:, ...
471:111111
369:, and
245:(4â2)
187:(3+4)
161:(8+9)
44:, and
34:digits
20:is an
4512:Prime
4507:Lucky
4496:sieve
4425:Other
4411:Smith
4291:Digit
4249:Happy
4224:Keith
4197:Other
4041:Rough
4011:Giuga
3476:Euler
3338:Cubic
2992:Lucas
2888:Proth
501:121,
489:121,
467:11011
409:<
405:when
242:1024
227:6Ă21
216:1022
207:3Ă51
167:8Ă86
88:prime
4466:Evil
4146:Self
4096:and
3986:Blum
3697:and
3501:Lobb
3456:Cake
3451:Bell
3201:Star
3108:Ulam
3007:Pell
2796:Cube
2689:OEIS
2631:OEIS
2487:and
2447:and
2243:>
1359:1000
1346:1111
1321:1111
1181:as 2
833:>
819:1).
807:as 1
507:1242
454:base
421:and
353:and
296:OEIS
281:nice
277:nice
269:OEIS
253:nice
236:625
230:216
224:126
219:2â2
213:7+4
210:347
204:153
198:125
193:3+7
190:736
184:343
178:128
172:121
164:688
158:289
155:2â1
152:127
103:OEIS
65:OEIS
4584:Ban
3972:or
3491:Lah
2676:doi
2672:161
2641:in
2564:161
1450:101
1437:102
1412:102
661:,
612:16
605:336
600:15
593:144
589:135
584:14
576:13
568:12
560:11
553:314
530:144
503:224
491:123
389:or
175:11
146:25
4654::
2704:.
2700:.
2670:.
2666:.
2653:.
2599:^
2562:,
2317:10
2306:50
2302:10
2279:50
2275:10
2246:10
2176:10
2165:60
2161:10
2140:60
2136:10
2067:10
2056:60
2052:10
2031:60
2027:10
1958:10
1947:60
1943:10
1922:60
1918:10
1849:10
1838:60
1834:10
1813:60
1809:10
1779:10
1688:10
1667:13
1663:10
1632:11
1597:10
1576:13
1572:10
1531:33
1522:20
1518:10
1497:20
1493:10
1433:10
1408:10
1355:10
1342:10
1317:10
1235:.
1185:=
1041:,
848:,
815:Ă
811:=
809:mk
591:,
548:9
541:33
537:8
525:7
518:24
514:6
498:5
486:4
478:3
469:,
462:2
429:.
425:/
417:â
401:+
399:mb
393:Ă
385:+
377:=
373:+
371:mb
365:=
361:+
359:mb
341:,
329:+
327:mb
298:).
275:A
271:).
251:A
239:5
233:6
201:5
181:2
149:5
105:).
82:A
67:).
16:A
2856:a
2737:e
2730:t
2723:v
2684:.
2678::
2616:.
2593:.
2579:.
2501:1
2498:+
2495:b
2475:b
2455:q
2435:p
2415:q
2412:+
2409:n
2406:p
2386:n
2360:+
2357:0
2354:+
2351:0
2348:+
2343:5
2339:)
2334:A
2329:/
2324:A
2320:+
2314:(
2311:+
2295:n
2292:=
2284:+
2268:n
2240:b
2213:+
2210:0
2207:+
2204:0
2201:+
2196:5
2192:)
2188:0
2182:1
2179:+
2173:(
2170:+
2154:n
2151:=
2145:+
2129:n
2104:+
2101:0
2098:+
2095:0
2092:+
2087:5
2083:)
2079:1
2073:2
2070:+
2064:(
2061:+
2045:n
2042:=
2036:+
2020:n
1995:+
1992:0
1989:+
1986:0
1983:+
1978:5
1974:)
1970:2
1964:3
1961:+
1955:(
1952:+
1936:n
1933:=
1927:+
1911:n
1886:+
1883:0
1880:+
1877:0
1874:+
1869:5
1865:)
1861:3
1855:4
1852:+
1846:(
1843:+
1827:n
1824:=
1818:+
1802:n
1773:b
1767:7
1741:)
1738:4
1735:+
1732:3
1729:(
1725:)
1721:2
1718:+
1715:5
1712:(
1709:+
1704:1
1698:5
1692:2
1681:n
1678:=
1672:+
1656:n
1628:)
1624:3
1618:2
1615:(
1612:+
1607:4
1604:+
1601:4
1590:n
1587:=
1581:+
1565:n
1543:0
1540:+
1535:3
1527:+
1511:n
1508:=
1502:+
1486:n
1464:0
1461:+
1458:0
1455:+
1446:2
1442:+
1426:n
1423:=
1417:+
1401:n
1379:0
1376:+
1373:0
1370:+
1367:1
1351:+
1335:n
1332:=
1326:+
1310:n
1287:6
1281:b
1275:2
1215:5
1212:=
1209:k
1199:n
1195:k
1191:k
1187:k
1183:k
1169:b
1144:2
1140:k
1136:=
1133:k
1130:+
1127:k
1119:2
1115:k
1111:=
1108:k
1105:+
1101:)
1096:2
1092:)
1089:1
1083:k
1080:(
1077:k
1071:(
1067:2
1064:=
1061:k
1058:+
1055:b
1052:2
1027:2
1023:)
1020:1
1014:k
1011:(
1008:k
1002:=
999:b
976:1
970:n
950:b
927:1
924:+
918:n
914:b
909:2
906:+
901:n
898:2
894:b
890:=
885:2
880:)
877:1
874:+
869:n
865:b
861:(
836:2
830:b
817:m
813:k
795:b
772:)
769:1
766:+
763:b
760:m
757:(
754:k
751:=
748:k
745:+
742:k
739:b
736:m
733:=
730:k
727:+
724:b
721:)
718:m
715:+
712:m
706:k
703:m
700:(
697:=
694:k
691:+
688:b
685:m
682:+
677:2
673:b
649:m
643:k
640:m
637:=
634:b
439:2
427:n
423:m
419:n
415:m
411:b
407:n
403:n
395:n
391:m
387:n
383:m
379:n
375:n
367:m
363:n
355:n
351:m
347:b
343:n
339:m
335:b
331:n
308:n
304:n
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