4940:
2461:
It is not known whether all non-palindromic numbers can be paired with palindromic numbers in this way. While no number has been proven to be unpaired, many do not appear to be. For example, 196 does not yield a palindrome even after 700,000,000 iterations. Any number that never becomes palindromic
2457:
Non-palindromic numbers can be paired with palindromic ones via a series of operations. First, the non-palindromic number is reversed and the result is added to the original number. If the result is not a palindromic number, this is repeated until it gives a palindromic number. Such number is called
2547:
must be at least 13, the primorial must be at least 1·2·3·5·7·11·13, and 7×11×13 = 1001. Fuller also refers to powers of 1001 as
Scheherazade numbers. The smallest primorial containing Scheherazade number is 13# = 30,030.
2652:
on a separate line. If this is done, using more spillover lines as necessary, the symmetry is preserved indefinitely to any power. Many other
Scheherazade numbers show similar symmetries when expressed in this way.
360:
There are likewise 90 palindromic numbers with four digits (again, 9 choices for the first digit multiplied by ten choices for the second digit. The other two digits are determined by the choice of the first two):
1740:
2473:
and announced "The
Largest Known Most Delayed Palindrome". The sequence of 125 261-step most delayed palindromes preceding 1,999,291,987,030,606,810 and not reported before was published separately as
1975:. For example, the number 6 is written as "110" in base 2, "20" in base 3, and "12" in base 4, none of which are palindromes. All strictly non-palindromic numbers larger than 6 are prime. Indeed, if
1400:
The only known non-palindromic number whose cube is a palindrome is 2201, and it is a conjecture the fourth root of all the palindrome fourth powers are a palindrome with 100000...000001 (10 + 1).
2639:
248:
1954:
2551:
Fuller pointed out that some of these numbers are palindromic by groups of digits. For instance 17# = 510,510 shows a symmetry of groups of three digits. Fuller called such numbers
1786:
2447:
1872:
2661:
In 2018, a paper was published demonstrating that every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater.
2192:
2060:
1648:
2123:
1999:
1549:
2366:
2028:
2335:
2153:
2090:
1595:
2514:. Fuller does not give a formal definition for this term, but from the examples he gives, it can be understood to be those numbers that contain a factor of the
2386:
1912:
1892:
1826:
1806:
1615:
1569:
1523:
1503:
2902:
3042:
372:
There are 1099 palindromic numbers smaller than 10 and for other exponents of 10 we have: 1999, 10999, 19999, 109999, 199999, 1099999, ... (sequence
2830:
1653:
59:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, ... (sequence
2858:
4964:
2751:
2711:
2492:
2204:
1467:
1449:
1394:
1380:
1363:
1346:
379:
113:
96:
66:
1650:). Even excluding cases where the number is smaller than the base, most numbers are palindromic in more than one base. For example,
3035:
2485:
The sum of the reciprocals of the palindromic numbers is a convergent series, whose value is approximately 3.37028... (sequence
365:{1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, ..., 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999},
352:: 9 choices for the first digit - which determines the third digit as well - multiplied by 10 choices for the second digit):
2569:
3842:
3028:
3837:
2951:
1387:
The first nine terms of the sequence 1, 11, 111, 1111, ... form the palindromes 1, 121, 12321, 1234321, ... (sequence
3852:
2947:
2842:
3832:
4545:
4125:
2884:
189:
3847:
1917:
2918:
4631:
132:
of numbers written (in that base) as 101, 1001, 10001, 100001, etc. consists solely of palindromic numbers.
4297:
3947:
3616:
3409:
4473:
4332:
4163:
3977:
3967:
3621:
3601:
2539:
2302:, 311, 317, 347, 359, 367, 389, 439, 491, 563, 569, 593, 607, 659, 739, 827, 853, 877, 977, 983, 997, ...
356:{101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919, 929, 939, 949, 959, 969, 979, 989, 999}
31:) is a number (such as 16461) that remains the same when its digits are reversed. In other words, it has
4302:
1745:
4422:
4045:
3887:
3802:
3611:
3593:
3487:
3477:
3467:
3303:
2394:
1831:
4327:
4550:
4095:
3716:
3502:
3497:
3492:
3482:
3459:
4307:
2311:
If the digits of a natural number don't only have to be reversed in order, but also subtracted from
3972:
3882:
3535:
2940:
A Number for Your
Thoughts: Facts and Speculations about Number from Euclid to the latest Computers
2825:
74:
2988:
2158:
2033:
51:) whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers (in
4661:
4626:
4412:
4322:
4196:
4171:
4080:
4070:
3792:
3682:
3664:
3584:
4921:
4191:
4065:
3696:
3472:
3252:
3179:
2804:
Dvorakova, Lubomira; Kruml, Stanislav; Ryzak, David (16 Aug 2020). "Antipalindromic numbers".
2728:
2559:
numbers that are palindromic in three-digit groups, but also the values of the groups are the
1620:
4885:
4525:
4176:
4030:
3957:
3112:
2095:
1435:
32:
1978:
1528:
4969:
4818:
4712:
4676:
4417:
4140:
4120:
3937:
3606:
3394:
2855:
2560:
2344:
466:
3897:
3366:
2678:
2004:
8:
4540:
4404:
4399:
4367:
4130:
4105:
4100:
4075:
4005:
4001:
3932:
3822:
3654:
3450:
3419:
2314:
2132:
2069:
1574:
627:
4943:
4697:
4692:
4580:
4478:
4457:
4229:
4110:
4060:
3982:
3952:
3892:
3659:
3639:
3570:
3283:
2969:
2892:
2805:
2786:
2645:
2507:
2371:
1897:
1877:
1811:
1791:
1600:
1554:
1508:
1488:
3827:
2999:
668:
382:). The number of palindromic numbers which have some other property are listed below:
4939:
4837:
4782:
4636:
4611:
4585:
4040:
4035:
3962:
3942:
3927:
3649:
3631:
3550:
3540:
3525:
3288:
2966:
2943:
2838:
1227:
86:
4362:
4873:
4666:
4252:
4224:
4214:
4206:
4090:
4055:
4050:
4017:
3711:
3674:
3565:
3560:
3555:
3545:
3517:
3404:
3351:
3308:
3247:
3010:
2778:
2469:
On
January 24, 2017, the number 1,999,291,987,030,606,810 was published in OEIS as
1403:
1268:
3356:
1442:
0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, ... (sequence
4849:
4738:
4671:
4597:
4520:
4494:
4312:
4025:
3817:
3787:
3777:
3772:
3438:
3346:
3293:
3137:
3077:
3005:
2983:
2922:
2862:
2834:
349:
172:
149:
2648:
taken into the group to the left in some groups. Fuller suggests writing these
2555:, or SSRCD numbers. Fuller notes that 1001 raised to a power not only produces
4854:
4722:
4707:
4571:
4535:
4510:
4386:
4357:
4342:
4219:
4115:
4085:
3812:
3767:
3644:
3242:
3237:
3232:
3204:
3189:
3102:
3087:
3065:
3052:
2463:
1479:
1427:
1353:
574:
426:
153:
1339:: 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ... (sequence
4958:
4777:
4761:
4702:
4656:
4352:
4337:
4247:
3530:
3399:
3361:
3318:
3199:
3184:
3174:
3132:
3122:
3097:
3020:
3014:
2993:
1336:
1317:
546:
103:
4813:
4802:
4717:
4555:
4530:
4447:
4347:
4317:
4292:
4276:
4181:
4148:
3871:
3782:
3721:
3298:
3194:
3127:
3107:
3082:
2534:
2526:
2341:. Formally, in the usual decomposition of a natural number into its digits
2299:
2295:
2291:
2287:
2283:
2279:
2275:
2271:
2267:
2263:
2259:
2255:
1475:
1370:
731:
599:
125:
4772:
4647:
4452:
3916:
3807:
3762:
3757:
3507:
3414:
3313:
3142:
3117:
3092:
2766:
2741:
2251:
2247:
2243:
2239:
2235:
342:
4909:
4890:
4186:
3797:
2984:
Jason
Doucette - 196 Palindrome Quest / Most Delayed Palindromic Number
2790:
2683:
2231:
2227:
2223:
2219:
2215:
2211:
506:
341:
All palindromic numbers with an even number of digits are divisible by
310:
40:
4515:
4442:
4434:
4239:
4153:
3271:
2974:
2515:
77:. A typical problem asks for numbers that possess a certain property
2782:
2337:
to yield the original sequence again, then the number is said to be
4616:
2915:
2897:
2810:
129:
1356:: 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ... (sequence
4621:
4280:
4274:
1735:{\displaystyle 1221_{4}=151_{8}=77_{14}=55_{20}=33_{34}=11_{104}}
1431:
322:
141:
52:
2543:, telling a new story each night to delay her execution. Since
970:
even squarefree with an even number of (distinct) prime factors
313:
is written 0 in any base and is also palindromic by definition.
2883:
Cilleruelo, Javier; Luca, Florian; Baxter, Lewis (2016-02-19).
2746:"Sequence A016038 (Strictly non-palindromic numbers)"
1438:
palindromic numbers are those with the binary representations:
1008:
odd squarefree with an even number of (distinct) prime factors
3336:
348:
There are 90 palindromic numbers with three digits (Using the
167: ≥ 2, where it is written in standard notation with
140:
Although palindromic numbers are most often considered in the
161:
121:
2725:
Problems in applied mathematics: selections from SIAM review
2964:
2745:
2706:
2553:
Scheherazade
Sublimely Rememberable Comprehensive Dividends
2487:
2474:
2470:
2199:
1462:
1444:
1389:
1375:
1358:
1341:
374:
108:
106:
are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... (sequence
91:
73:
Palindromic numbers receive most attention in the realm of
61:
2389:
2197:
The first few strictly non-palindromic numbers (sequence
16:
Number that remains the same when its digits are reversed
1460:
0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, ... (sequence
1373:: 0, 1, 14641, 104060401, 1004006004001, ... (sequence
2885:"Every positive integer is a sum of three palindromes"
2740:
369:
so there are 199 palindromic numbers smaller than 10.
4000:
2827:
Synergetics: Explorations in the
Geometry of thinking
2572:
2397:
2374:
2347:
2317:
2161:
2135:
2098:
2072:
2036:
2007:
1981:
1920:
1900:
1880:
1834:
1814:
1794:
1748:
1656:
1623:
1603:
1577:
1557:
1531:
1511:
1491:
192:
4385:
2882:
2634:{\displaystyle (1001)^{6}=1,006,015,020,015,006,001}
128:
palindromic numbers, since in any base the infinite
2803:
2714:) The next example is 19 digits - 900075181570009.
2633:
2441:
2380:
2360:
2329:
2186:
2147:
2117:
2084:
2054:
2022:
1993:
1948:
1906:
1886:
1866:
1820:
1800:
1780:
1734:
1642:
1609:
1589:
1563:
1543:
1517:
1497:
856:even with an odd number of distinct prime factors
325:there are ten palindromic numbers with one digit:
321:All numbers with one digit are palindromic, so in
242:
3384:
2644:This sequence fails at (1001) because there is a
1571:is then a single-digit number), and also in base
932:odd with an odd number of distinct prime factors
333:There are 9 palindromic numbers with two digits:
89:are 2, 3, 5, 7, 11, 101, 131, 151, ... (sequence
4956:
1482:form a subset of the binary palindromic primes.
776:with an odd number of distinct prime factors (μ(
3270:
3064:
3050:
2824:R. Buckminster Fuller, with E. J. Applewhite,
1959:A number that is non-palindromic in all bases
316:
3036:
2769:(1989). "Conway's RATS and other reversals".
1406:conjectured there are no palindromes of form
4872:
3222:
2529:in the number. Fuller called these numbers
1151:even with exactly 3 distinct prime factors
243:{\displaystyle n=\sum _{i=0}^{k}a_{i}b^{i}}
3337:Possessing a specific set of other numbers
3160:
3043:
3029:
2533:because they must have a factor of 1001.
2306:
4800:
3747:
3015:Numerical Palindromes and the 196 Problem
2896:
2809:
2752:On-Line Encyclopedia of Integer Sequences
2480:
1949:{\displaystyle {\sqrt {p}}\leq b\leq p-2}
1426:Palindromic numbers can be considered in
818:even with an odd number of prime factors
894:odd with an odd number of prime factors
2679:"The Prime Glossary: palindromic prime"
2498:
4957:
4908:
2656:
4907:
4871:
4835:
4799:
4759:
4384:
4273:
3999:
3914:
3869:
3746:
3436:
3383:
3335:
3269:
3221:
3159:
3063:
3024:
3017:, IJPAM, Vol.80, No.3, 375–384, 2012.
2965:
2952:Limited Online-Version (Google Books)
1971: − 2 can be called a
337:{11, 22, 33, 44, 55, 66, 77, 88, 99}.
3437:
135:
4836:
2765:
2506:are a set of numbers identified by
13:
4760:
2452:
1781:{\displaystyle 1991_{10}=7C7_{16}}
1113:even with exactly 3 prime factors
1084:even with exactly 2 prime factors
43:, which refers to a word (such as
14:
4981:
2958:
2771:The American Mathematical Monthly
2442:{\displaystyle a_{i}=b-1-a_{k-i}}
2092:, or else it is a perfect square
1867:{\displaystyle n/2\leq b\leq n-2}
1311:
1189:odd with exactly 3 prime factors
1046:odd with exactly 2 prime factors
35:across a vertical axis. The term
4965:Base-dependent integer sequences
4938:
4546:Perfect digit-to-digit invariant
3915:
2155:(except for the special case of
2129:is the palindrome "121" in base
730:with an even number of distinct
2905:from the original on 2021-02-12
2876:
2066:is the palindrome "aa" in base
1973:strictly non-palindromic number
329:{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
81:are palindromic. For instance:
3006:P. De Geest, Palindromic cubes
3000:Palindromic Numbers to 100,000
2994:On General Palindromic Numbers
2867:
2848:
2818:
2797:
2759:
2734:
2717:
2698:
2671:
2580:
2573:
2388:, a number is antipalindromic
1421:
283:is palindromic if and only if
253:with, as usual, 0 ≤
1:
3385:Expressible via specific sums
2989:196 and Other Lychrel Numbers
2932:
1894:is never palindromic in base
1808:is never palindromic in base
2187:{\displaystyle n=9=1001_{2}}
2055:{\displaystyle 2\leq a<b}
1505:is palindromic in all bases
7:
4474:Multiplicative digital root
2540:One Thousand and One Nights
1963:in the range 2 ≤
1874:. Moreover, a prime number
1316:There are many palindromic
317:Decimal palindromic numbers
10:
4986:
3870:
2889:Mathematics of Computation
2742:Sloane, N. J. A.
2723:Murray S. Klamkin (1990),
2462:in this way is known as a
2001:is composite, then either
120:It is obvious that in any
4934:
4917:
4903:
4881:
4867:
4845:
4831:
4809:
4795:
4768:
4755:
4731:
4685:
4645:
4596:
4570:
4551:Perfect digital invariant
4503:
4487:
4466:
4433:
4398:
4394:
4380:
4288:
4269:
4238:
4205:
4162:
4139:
4126:Superior highly composite
4016:
4012:
3995:
3923:
3910:
3878:
3865:
3753:
3742:
3704:
3695:
3673:
3630:
3592:
3583:
3516:
3458:
3449:
3445:
3432:
3390:
3379:
3342:
3331:
3279:
3265:
3228:
3217:
3170:
3155:
3073:
3059:
1098:
1095:
1092:
721:
718:
715:
618:
615:
612:
609:
587:
584:
578:
565:
562:
553:
550:
4164:Euler's totient function
3948:Euler–Jacobi pseudoprime
3223:Other polynomial numbers
2664:
2458:"a delayed palindrome".
1643:{\displaystyle 11_{n-1}}
1327:is a natural number and
75:recreational mathematics
3978:Somer–Lucas pseudoprime
3968:Lucas–Carmichael number
3803:Lazy caterer's sequence
2525:≥13 and is the largest
2307:Antipalindromic numbers
2118:{\displaystyle n=a^{2}}
1551:(trivially so, because
713:square with prime root
144:system, the concept of
3853:Wedderburn–Etherington
3253:Lucky numbers of Euler
2635:
2557:sublimely rememberable
2537:is the storyteller of
2481:Sum of the reciprocals
2443:
2382:
2362:
2331:
2188:
2149:
2119:
2086:
2056:
2024:
1995:
1994:{\displaystyle n>6}
1950:
1908:
1888:
1868:
1822:
1802:
1782:
1736:
1644:
1611:
1591:
1565:
1545:
1544:{\displaystyle b>n}
1519:
1499:
244:
219:
148:can be applied to the
4141:Prime omega functions
3958:Frobenius pseudoprime
3748:Combinatorial numbers
3617:Centered dodecahedral
3410:Primary pseudoperfect
2636:
2561:binomial coefficients
2444:
2383:
2363:
2361:{\displaystyle a_{i}}
2332:
2189:
2150:
2120:
2087:
2057:
2025:
1996:
1951:
1909:
1889:
1869:
1823:
1803:
1783:
1737:
1645:
1612:
1592:
1566:
1546:
1520:
1500:
279: ≠ 0. Then
245:
199:
160: > 0 in
33:reflectional symmetry
4600:-composition related
4400:Arithmetic functions
4002:Arithmetic functions
3938:Elliptic pseudoprime
3622:Centered icosahedral
3602:Centered tetrahedral
2970:"Palindromic Number"
2570:
2531:Scheherazade numbers
2504:Scheherazade numbers
2499:Scheherazade numbers
2395:
2372:
2345:
2315:
2159:
2133:
2096:
2070:
2034:
2023:{\displaystyle n=ab}
2005:
1979:
1918:
1898:
1878:
1832:
1812:
1792:
1746:
1654:
1621:
1601:
1575:
1555:
1529:
1509:
1489:
190:
156:. Consider a number
4526:Kaprekar's constant
4046:Colossally abundant
3933:Catalan pseudoprime
3833:Schröder–Hipparchus
3612:Centered octahedral
3488:Centered heptagonal
3478:Centered pentagonal
3468:Centered triangular
3068:and related numbers
2873:Fuller, pp. 777-780
2657:Sums of palindromes
2330:{\displaystyle b-1}
2148:{\displaystyle a-1}
2085:{\displaystyle b-1}
1590:{\displaystyle n-1}
1434:. For example, the
4944:Mathematics portal
4886:Aronson's sequence
4632:Smarandache–Wellin
4389:-dependent numbers
4096:Primitive abundant
3983:Strong pseudoprime
3973:Perrin pseudoprime
3953:Fermat pseudoprime
3893:Wolstenholme prime
3717:Squared triangular
3503:Centered decagonal
3498:Centered nonagonal
3493:Centered octagonal
3483:Centered hexagonal
2967:Weisstein, Eric W.
2942:: CRC Press 1986,
2938:Malcolm E. Lines:
2921:2019-02-08 at the
2861:2016-03-05 at the
2837:, Macmillan, 1982
2833:2016-02-27 at the
2755:. OEIS Foundation.
2631:
2508:Buckminster Fuller
2439:
2378:
2358:
2327:
2184:
2145:
2115:
2082:
2052:
2020:
1991:
1946:
1904:
1884:
1864:
1818:
1798:
1778:
1732:
1640:
1607:
1587:
1561:
1541:
1515:
1495:
240:
87:palindromic primes
29:numeric palindrome
25:numeral palindrome
21:palindromic number
4952:
4951:
4930:
4929:
4899:
4898:
4863:
4862:
4827:
4826:
4791:
4790:
4751:
4750:
4747:
4746:
4566:
4565:
4376:
4375:
4265:
4264:
4261:
4260:
4207:Aliquot sequences
4018:Divisor functions
3991:
3990:
3963:Lucas pseudoprime
3943:Euler pseudoprime
3928:Carmichael number
3906:
3905:
3861:
3860:
3738:
3737:
3734:
3733:
3730:
3729:
3691:
3690:
3579:
3578:
3536:Square triangular
3428:
3427:
3375:
3374:
3327:
3326:
3261:
3260:
3213:
3212:
3151:
3150:
3002:from Ask Dr. Math
2381:{\displaystyle b}
1926:
1907:{\displaystyle b}
1887:{\displaystyle p}
1821:{\displaystyle b}
1801:{\displaystyle n}
1610:{\displaystyle n}
1564:{\displaystyle n}
1518:{\displaystyle b}
1498:{\displaystyle n}
1309:
1308:
1228:Carmichael number
136:Formal definition
23:(also known as a
4977:
4942:
4905:
4904:
4874:Natural language
4869:
4868:
4833:
4832:
4801:Generated via a
4797:
4796:
4757:
4756:
4662:Digit-reassembly
4627:Self-descriptive
4431:
4430:
4396:
4395:
4382:
4381:
4333:Lucas–Carmichael
4323:Harmonic divisor
4271:
4270:
4197:Sparsely totient
4172:Highly cototient
4081:Multiply perfect
4071:Highly composite
4014:
4013:
3997:
3996:
3912:
3911:
3867:
3866:
3848:Telephone number
3744:
3743:
3702:
3701:
3683:Square pyramidal
3665:Stella octangula
3590:
3589:
3456:
3455:
3447:
3446:
3439:Figurate numbers
3434:
3433:
3381:
3380:
3333:
3332:
3267:
3266:
3219:
3218:
3157:
3156:
3061:
3060:
3045:
3038:
3031:
3022:
3021:
3011:Yutaka Nishiyama
2980:
2979:
2926:
2913:
2911:
2910:
2900:
2880:
2874:
2871:
2865:
2852:
2846:
2822:
2816:
2815:
2813:
2801:
2795:
2794:
2763:
2757:
2756:
2738:
2732:
2721:
2715:
2709:
2702:
2696:
2695:
2693:
2691:
2675:
2640:
2638:
2637:
2632:
2588:
2587:
2563:. For instance,
2490:
2448:
2446:
2445:
2440:
2438:
2437:
2407:
2406:
2387:
2385:
2384:
2379:
2367:
2365:
2364:
2359:
2357:
2356:
2336:
2334:
2333:
2328:
2202:
2193:
2191:
2190:
2185:
2183:
2182:
2154:
2152:
2151:
2146:
2125:, in which case
2124:
2122:
2121:
2116:
2114:
2113:
2091:
2089:
2088:
2083:
2062:, in which case
2061:
2059:
2058:
2053:
2029:
2027:
2026:
2021:
2000:
1998:
1997:
1992:
1955:
1953:
1952:
1947:
1927:
1922:
1913:
1911:
1910:
1905:
1893:
1891:
1890:
1885:
1873:
1871:
1870:
1865:
1842:
1827:
1825:
1824:
1819:
1807:
1805:
1804:
1799:
1787:
1785:
1784:
1779:
1777:
1776:
1758:
1757:
1741:
1739:
1738:
1733:
1731:
1730:
1718:
1717:
1705:
1704:
1692:
1691:
1679:
1678:
1666:
1665:
1649:
1647:
1646:
1641:
1639:
1638:
1616:
1614:
1613:
1608:
1596:
1594:
1593:
1588:
1570:
1568:
1567:
1562:
1550:
1548:
1547:
1542:
1524:
1522:
1521:
1516:
1504:
1502:
1501:
1496:
1465:
1447:
1404:Gustavus Simmons
1392:
1378:
1361:
1344:
667:non-squarefree (
385:
384:
377:
262: <
249:
247:
246:
241:
239:
238:
229:
228:
218:
213:
111:
102:The palindromic
94:
64:
39:is derived from
4985:
4984:
4980:
4979:
4978:
4976:
4975:
4974:
4955:
4954:
4953:
4948:
4926:
4922:Strobogrammatic
4913:
4895:
4877:
4859:
4841:
4823:
4805:
4787:
4764:
4743:
4727:
4686:Divisor-related
4681:
4641:
4592:
4562:
4499:
4483:
4462:
4429:
4402:
4390:
4372:
4284:
4283:related numbers
4257:
4234:
4201:
4192:Perfect totient
4158:
4135:
4066:Highly abundant
4008:
3987:
3919:
3902:
3874:
3857:
3843:Stirling second
3749:
3726:
3687:
3669:
3626:
3575:
3512:
3473:Centered square
3441:
3424:
3386:
3371:
3338:
3323:
3275:
3274:defined numbers
3257:
3224:
3209:
3180:Double Mersenne
3166:
3147:
3069:
3055:
3053:natural numbers
3049:
2961:
2935:
2930:
2929:
2923:Wayback Machine
2908:
2906:
2881:
2877:
2872:
2868:
2863:Wayback Machine
2853:
2849:
2835:Wayback Machine
2823:
2819:
2802:
2798:
2783:10.2307/2325149
2767:Guy, Richard K.
2764:
2760:
2739:
2735:
2722:
2718:
2705:
2703:
2699:
2689:
2687:
2677:
2676:
2672:
2667:
2659:
2583:
2579:
2571:
2568:
2567:
2501:
2486:
2483:
2455:
2453:Lychrel process
2427:
2423:
2402:
2398:
2396:
2393:
2392:
2373:
2370:
2369:
2352:
2348:
2346:
2343:
2342:
2339:antipalindromic
2316:
2313:
2312:
2309:
2198:
2178:
2174:
2160:
2157:
2156:
2134:
2131:
2130:
2109:
2105:
2097:
2094:
2093:
2071:
2068:
2067:
2035:
2032:
2031:
2006:
2003:
2002:
1980:
1977:
1976:
1921:
1919:
1916:
1915:
1899:
1896:
1895:
1879:
1876:
1875:
1838:
1833:
1830:
1829:
1813:
1810:
1809:
1793:
1790:
1789:
1772:
1768:
1753:
1749:
1747:
1744:
1743:
1726:
1722:
1713:
1709:
1700:
1696:
1687:
1683:
1674:
1670:
1661:
1657:
1655:
1652:
1651:
1628:
1624:
1622:
1619:
1618:
1602:
1599:
1598:
1576:
1573:
1572:
1556:
1553:
1552:
1530:
1527:
1526:
1510:
1507:
1506:
1490:
1487:
1486:
1480:Mersenne primes
1461:
1456:or in decimal:
1443:
1428:numeral systems
1424:
1388:
1374:
1357:
1340:
1314:
1275:is palindromic
373:
350:rule of product
319:
304:
291:
278:
261:
234:
230:
224:
220:
214:
203:
191:
188:
187:
182:
150:natural numbers
138:
126:infinitely many
107:
90:
60:
17:
12:
11:
5:
4983:
4973:
4972:
4967:
4950:
4949:
4947:
4946:
4935:
4932:
4931:
4928:
4927:
4925:
4924:
4918:
4915:
4914:
4901:
4900:
4897:
4896:
4894:
4893:
4888:
4882:
4879:
4878:
4865:
4864:
4861:
4860:
4858:
4857:
4855:Sorting number
4852:
4850:Pancake number
4846:
4843:
4842:
4829:
4828:
4825:
4824:
4822:
4821:
4816:
4810:
4807:
4806:
4793:
4792:
4789:
4788:
4786:
4785:
4780:
4775:
4769:
4766:
4765:
4762:Binary numbers
4753:
4752:
4749:
4748:
4745:
4744:
4742:
4741:
4735:
4733:
4729:
4728:
4726:
4725:
4720:
4715:
4710:
4705:
4700:
4695:
4689:
4687:
4683:
4682:
4680:
4679:
4674:
4669:
4664:
4659:
4653:
4651:
4643:
4642:
4640:
4639:
4634:
4629:
4624:
4619:
4614:
4609:
4603:
4601:
4594:
4593:
4591:
4590:
4589:
4588:
4577:
4575:
4572:P-adic numbers
4568:
4567:
4564:
4563:
4561:
4560:
4559:
4558:
4548:
4543:
4538:
4533:
4528:
4523:
4518:
4513:
4507:
4505:
4501:
4500:
4498:
4497:
4491:
4489:
4488:Coding-related
4485:
4484:
4482:
4481:
4476:
4470:
4468:
4464:
4463:
4461:
4460:
4455:
4450:
4445:
4439:
4437:
4428:
4427:
4426:
4425:
4423:Multiplicative
4420:
4409:
4407:
4392:
4391:
4387:Numeral system
4378:
4377:
4374:
4373:
4371:
4370:
4365:
4360:
4355:
4350:
4345:
4340:
4335:
4330:
4325:
4320:
4315:
4310:
4305:
4300:
4295:
4289:
4286:
4285:
4267:
4266:
4263:
4262:
4259:
4258:
4256:
4255:
4250:
4244:
4242:
4236:
4235:
4233:
4232:
4227:
4222:
4217:
4211:
4209:
4203:
4202:
4200:
4199:
4194:
4189:
4184:
4179:
4177:Highly totient
4174:
4168:
4166:
4160:
4159:
4157:
4156:
4151:
4145:
4143:
4137:
4136:
4134:
4133:
4128:
4123:
4118:
4113:
4108:
4103:
4098:
4093:
4088:
4083:
4078:
4073:
4068:
4063:
4058:
4053:
4048:
4043:
4038:
4033:
4031:Almost perfect
4028:
4022:
4020:
4010:
4009:
3993:
3992:
3989:
3988:
3986:
3985:
3980:
3975:
3970:
3965:
3960:
3955:
3950:
3945:
3940:
3935:
3930:
3924:
3921:
3920:
3908:
3907:
3904:
3903:
3901:
3900:
3895:
3890:
3885:
3879:
3876:
3875:
3863:
3862:
3859:
3858:
3856:
3855:
3850:
3845:
3840:
3838:Stirling first
3835:
3830:
3825:
3820:
3815:
3810:
3805:
3800:
3795:
3790:
3785:
3780:
3775:
3770:
3765:
3760:
3754:
3751:
3750:
3740:
3739:
3736:
3735:
3732:
3731:
3728:
3727:
3725:
3724:
3719:
3714:
3708:
3706:
3699:
3693:
3692:
3689:
3688:
3686:
3685:
3679:
3677:
3671:
3670:
3668:
3667:
3662:
3657:
3652:
3647:
3642:
3636:
3634:
3628:
3627:
3625:
3624:
3619:
3614:
3609:
3604:
3598:
3596:
3587:
3581:
3580:
3577:
3576:
3574:
3573:
3568:
3563:
3558:
3553:
3548:
3543:
3538:
3533:
3528:
3522:
3520:
3514:
3513:
3511:
3510:
3505:
3500:
3495:
3490:
3485:
3480:
3475:
3470:
3464:
3462:
3453:
3443:
3442:
3430:
3429:
3426:
3425:
3423:
3422:
3417:
3412:
3407:
3402:
3397:
3391:
3388:
3387:
3377:
3376:
3373:
3372:
3370:
3369:
3364:
3359:
3354:
3349:
3343:
3340:
3339:
3329:
3328:
3325:
3324:
3322:
3321:
3316:
3311:
3306:
3301:
3296:
3291:
3286:
3280:
3277:
3276:
3263:
3262:
3259:
3258:
3256:
3255:
3250:
3245:
3240:
3235:
3229:
3226:
3225:
3215:
3214:
3211:
3210:
3208:
3207:
3202:
3197:
3192:
3187:
3182:
3177:
3171:
3168:
3167:
3153:
3152:
3149:
3148:
3146:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3095:
3090:
3085:
3080:
3074:
3071:
3070:
3057:
3056:
3048:
3047:
3040:
3033:
3025:
3019:
3018:
3008:
3003:
2997:
2991:
2986:
2981:
2960:
2959:External links
2957:
2956:
2955:
2934:
2931:
2928:
2927:
2916:arXiv preprint
2875:
2866:
2847:
2817:
2796:
2777:(5): 425–428.
2758:
2733:
2716:
2697:
2669:
2668:
2666:
2663:
2658:
2655:
2642:
2641:
2630:
2627:
2624:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2586:
2582:
2578:
2575:
2500:
2497:
2482:
2479:
2464:Lychrel number
2454:
2451:
2436:
2433:
2430:
2426:
2422:
2419:
2416:
2413:
2410:
2405:
2401:
2377:
2355:
2351:
2326:
2323:
2320:
2308:
2305:
2304:
2303:
2181:
2177:
2173:
2170:
2167:
2164:
2144:
2141:
2138:
2112:
2108:
2104:
2101:
2081:
2078:
2075:
2051:
2048:
2045:
2042:
2039:
2019:
2016:
2013:
2010:
1990:
1987:
1984:
1945:
1942:
1939:
1936:
1933:
1930:
1925:
1903:
1883:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1841:
1837:
1817:
1797:
1775:
1771:
1767:
1764:
1761:
1756:
1752:
1729:
1725:
1721:
1716:
1712:
1708:
1703:
1699:
1695:
1690:
1686:
1682:
1677:
1673:
1669:
1664:
1660:
1637:
1634:
1631:
1627:
1606:
1586:
1583:
1580:
1560:
1540:
1537:
1534:
1514:
1494:
1472:
1471:
1454:
1453:
1423:
1420:
1385:
1384:
1367:
1350:
1331:is 2, 3 or 4.
1318:perfect powers
1313:
1312:Perfect powers
1310:
1307:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1261:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1221:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1183:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1145:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1107:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1078:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1040:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1002:
1001:
998:
995:
992:
989:
986:
983:
980:
977:
974:
971:
964:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
926:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
888:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
857:
850:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
812:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
781:
770:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
724:
723:
720:
717:
714:
707:
706:
703:
700:
697:
694:
691:
688:
685:
682:
679:
676:
661:
660:
657:
654:
651:
648:
645:
642:
639:
636:
633:
630:
621:
620:
617:
614:
611:
608:
605:
602:
593:
592:
589:
586:
583:
580:
577:
568:
567:
564:
561:
558:
555:
552:
549:
540:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
500:
499:
496:
493:
490:
487:
484:
481:
478:
475:
472:
469:
460:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
420:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
367:
366:
358:
357:
339:
338:
331:
330:
318:
315:
296:
287:
274:
257:
251:
250:
237:
233:
227:
223:
217:
212:
209:
206:
202:
198:
195:
178:
154:numeral system
146:palindromicity
137:
134:
118:
117:
104:square numbers
100:
71:
70:
15:
9:
6:
4:
3:
2:
4982:
4971:
4968:
4966:
4963:
4962:
4960:
4945:
4941:
4937:
4936:
4933:
4923:
4920:
4919:
4916:
4911:
4906:
4902:
4892:
4889:
4887:
4884:
4883:
4880:
4875:
4870:
4866:
4856:
4853:
4851:
4848:
4847:
4844:
4839:
4834:
4830:
4820:
4817:
4815:
4812:
4811:
4808:
4804:
4798:
4794:
4784:
4781:
4779:
4776:
4774:
4771:
4770:
4767:
4763:
4758:
4754:
4740:
4737:
4736:
4734:
4730:
4724:
4721:
4719:
4716:
4714:
4713:Polydivisible
4711:
4709:
4706:
4704:
4701:
4699:
4696:
4694:
4691:
4690:
4688:
4684:
4678:
4675:
4673:
4670:
4668:
4665:
4663:
4660:
4658:
4655:
4654:
4652:
4649:
4644:
4638:
4635:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4608:
4605:
4604:
4602:
4599:
4595:
4587:
4584:
4583:
4582:
4579:
4578:
4576:
4573:
4569:
4557:
4554:
4553:
4552:
4549:
4547:
4544:
4542:
4539:
4537:
4534:
4532:
4529:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4509:
4508:
4506:
4502:
4496:
4493:
4492:
4490:
4486:
4480:
4477:
4475:
4472:
4471:
4469:
4467:Digit product
4465:
4459:
4456:
4454:
4451:
4449:
4446:
4444:
4441:
4440:
4438:
4436:
4432:
4424:
4421:
4419:
4416:
4415:
4414:
4411:
4410:
4408:
4406:
4401:
4397:
4393:
4388:
4383:
4379:
4369:
4366:
4364:
4361:
4359:
4356:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4303:Erdős–Nicolas
4301:
4299:
4296:
4294:
4291:
4290:
4287:
4282:
4278:
4272:
4268:
4254:
4251:
4249:
4246:
4245:
4243:
4241:
4237:
4231:
4228:
4226:
4223:
4221:
4218:
4216:
4213:
4212:
4210:
4208:
4204:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4169:
4167:
4165:
4161:
4155:
4152:
4150:
4147:
4146:
4144:
4142:
4138:
4132:
4129:
4127:
4124:
4122:
4121:Superabundant
4119:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4087:
4084:
4082:
4079:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4054:
4052:
4049:
4047:
4044:
4042:
4039:
4037:
4034:
4032:
4029:
4027:
4024:
4023:
4021:
4019:
4015:
4011:
4007:
4003:
3998:
3994:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3959:
3956:
3954:
3951:
3949:
3946:
3944:
3941:
3939:
3936:
3934:
3931:
3929:
3926:
3925:
3922:
3918:
3913:
3909:
3899:
3896:
3894:
3891:
3889:
3886:
3884:
3881:
3880:
3877:
3873:
3868:
3864:
3854:
3851:
3849:
3846:
3844:
3841:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3755:
3752:
3745:
3741:
3723:
3720:
3718:
3715:
3713:
3710:
3709:
3707:
3703:
3700:
3698:
3697:4-dimensional
3694:
3684:
3681:
3680:
3678:
3676:
3672:
3666:
3663:
3661:
3658:
3656:
3653:
3651:
3648:
3646:
3643:
3641:
3638:
3637:
3635:
3633:
3629:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3607:Centered cube
3605:
3603:
3600:
3599:
3597:
3595:
3591:
3588:
3586:
3585:3-dimensional
3582:
3572:
3569:
3567:
3564:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3544:
3542:
3539:
3537:
3534:
3532:
3529:
3527:
3524:
3523:
3521:
3519:
3515:
3509:
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3466:
3465:
3463:
3461:
3457:
3454:
3452:
3451:2-dimensional
3448:
3444:
3440:
3435:
3431:
3421:
3418:
3416:
3413:
3411:
3408:
3406:
3403:
3401:
3398:
3396:
3395:Nonhypotenuse
3393:
3392:
3389:
3382:
3378:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3344:
3341:
3334:
3330:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3295:
3292:
3290:
3287:
3285:
3282:
3281:
3278:
3273:
3268:
3264:
3254:
3251:
3249:
3246:
3244:
3241:
3239:
3236:
3234:
3231:
3230:
3227:
3220:
3216:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3172:
3169:
3164:
3158:
3154:
3144:
3141:
3139:
3136:
3134:
3133:Perfect power
3131:
3129:
3126:
3124:
3123:Seventh power
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3084:
3081:
3079:
3076:
3075:
3072:
3067:
3062:
3058:
3054:
3046:
3041:
3039:
3034:
3032:
3027:
3026:
3023:
3016:
3012:
3009:
3007:
3004:
3001:
2998:
2995:
2992:
2990:
2987:
2985:
2982:
2977:
2976:
2971:
2968:
2963:
2962:
2953:
2949:
2948:0-85274-495-1
2945:
2941:
2937:
2936:
2924:
2920:
2917:
2904:
2899:
2894:
2890:
2886:
2879:
2870:
2864:
2860:
2857:
2851:
2844:
2843:0-02-065320-4
2840:
2836:
2832:
2829:
2828:
2821:
2812:
2807:
2800:
2792:
2788:
2784:
2780:
2776:
2772:
2768:
2762:
2754:
2753:
2747:
2743:
2737:
2730:
2726:
2720:
2713:
2708:
2701:
2686:
2685:
2680:
2674:
2670:
2662:
2654:
2651:
2647:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2584:
2576:
2566:
2565:
2564:
2562:
2558:
2554:
2549:
2546:
2542:
2541:
2536:
2532:
2528:
2524:
2520:
2517:
2513:
2509:
2505:
2496:
2494:
2489:
2478:
2476:
2472:
2467:
2465:
2459:
2450:
2434:
2431:
2428:
2424:
2420:
2417:
2414:
2411:
2408:
2403:
2399:
2391:
2375:
2353:
2349:
2340:
2324:
2321:
2318:
2301:
2297:
2293:
2289:
2285:
2281:
2277:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2233:
2229:
2225:
2221:
2217:
2213:
2210:
2209:
2208:
2206:
2201:
2195:
2179:
2175:
2171:
2168:
2165:
2162:
2142:
2139:
2136:
2128:
2110:
2106:
2102:
2099:
2079:
2076:
2073:
2065:
2049:
2046:
2043:
2040:
2037:
2017:
2014:
2011:
2008:
1988:
1985:
1982:
1974:
1970:
1967: ≤
1966:
1962:
1957:
1943:
1940:
1937:
1934:
1931:
1928:
1923:
1901:
1881:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1839:
1835:
1815:
1795:
1773:
1769:
1765:
1762:
1759:
1754:
1750:
1727:
1723:
1719:
1714:
1710:
1706:
1701:
1697:
1693:
1688:
1684:
1680:
1675:
1671:
1667:
1662:
1658:
1635:
1632:
1629:
1625:
1604:
1584:
1581:
1578:
1558:
1538:
1535:
1532:
1512:
1492:
1483:
1481:
1477:
1476:Fermat primes
1469:
1464:
1459:
1458:
1457:
1451:
1446:
1441:
1440:
1439:
1437:
1433:
1429:
1419:
1417:
1413:
1409:
1405:
1401:
1398:
1396:
1391:
1382:
1377:
1372:
1371:fourth powers
1368:
1365:
1360:
1355:
1351:
1348:
1343:
1338:
1334:
1333:
1332:
1330:
1326:
1322:
1319:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1272:
1266:
1263:
1262:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1229:
1226:
1223:
1222:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1184:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1146:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1108:
1104:
1101:
1089:
1086:
1083:
1080:
1079:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1041:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1003:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
965:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
927:
923:
920:
917:
914:
911:
908:
905:
902:
899:
896:
893:
890:
889:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
851:
847:
844:
841:
838:
835:
832:
829:
826:
823:
820:
817:
814:
813:
809:
806:
803:
800:
797:
794:
791:
788:
785:
782:
779:
775:
772:
771:
767:
764:
761:
758:
755:
752:
749:
746:
743:
740:
737:
733:
732:prime factors
729:
726:
725:
712:
709:
708:
704:
701:
698:
695:
692:
689:
686:
683:
680:
677:
674:
672:
666:
663:
662:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
629:
626:
623:
622:
606:
603:
601:
598:
595:
594:
590:
581:
576:
573:
570:
569:
559:
556:
548:
545:
542:
541:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
508:
505:
502:
501:
497:
494:
491:
488:
485:
482:
479:
476:
473:
470:
468:
465:
462:
461:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
428:
425:
422:
421:
417:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
386:
383:
381:
376:
370:
364:
363:
362:
355:
354:
353:
351:
346:
344:
336:
335:
334:
328:
327:
326:
324:
314:
312:
308:
303:
299:
295:
292: =
290:
286:
282:
277:
273:
269:
265:
260:
256:
235:
231:
225:
221:
215:
210:
207:
204:
200:
196:
193:
186:
185:
184:
181:
177:
174:
170:
166:
163:
159:
155:
151:
147:
143:
133:
131:
127:
123:
115:
110:
105:
101:
98:
93:
88:
84:
83:
82:
80:
76:
68:
63:
58:
57:
56:
54:
50:
46:
42:
38:
34:
30:
26:
22:
4677:Transposable
4606:
4541:Narcissistic
4448:Digital root
4368:Super-Poulet
4328:Jordan–Pólya
4277:prime factor
4182:Noncototient
4149:Almost prime
4131:Superperfect
4106:Refactorable
4101:Quasiperfect
4076:Hyperperfect
3917:Pseudoprimes
3888:Wall–Sun–Sun
3823:Ordered Bell
3793:Fuss–Catalan
3705:non-centered
3655:Dodecahedral
3632:non-centered
3518:non-centered
3420:Wolstenholme
3165:× 2 ± 1
3162:
3161:Of the form
3128:Eighth power
3108:Fourth power
2996:at MathPages
2973:
2939:
2907:. Retrieved
2888:
2878:
2869:
2850:
2826:
2820:
2799:
2774:
2770:
2761:
2749:
2736:
2724:
2719:
2700:
2688:. Retrieved
2682:
2673:
2660:
2649:
2643:
2556:
2552:
2550:
2544:
2538:
2535:Scheherazade
2530:
2527:prime factor
2522:
2518:
2511:
2510:in his book
2503:
2502:
2484:
2468:
2460:
2456:
2338:
2310:
2196:
2126:
2063:
1972:
1968:
1964:
1960:
1958:
1484:
1473:
1455:
1425:
1415:
1414:> 4 (and
1411:
1407:
1402:
1399:
1386:
1369:Palindromic
1352:Palindromic
1335:Palindromic
1328:
1324:
1320:
1315:
1270:
1264:
1224:
1186:
1148:
1110:
1081:
1043:
1005:
967:
929:
891:
853:
815:
777:
773:
735:
727:
710:
670:
664:
624:
596:
571:
543:
503:
463:
423:
371:
368:
359:
347:
340:
332:
320:
306:
301:
297:
293:
288:
284:
280:
275:
271:
267:
263:
258:
254:
252:
179:
175:
168:
164:
157:
145:
139:
119:
78:
72:
48:
44:
36:
28:
24:
20:
18:
4970:Palindromes
4698:Extravagant
4693:Equidigital
4648:permutation
4607:Palindromic
4581:Automorphic
4479:Sum-product
4458:Sum-product
4413:Persistence
4308:Erdős–Woods
4230:Untouchable
4111:Semiperfect
4061:Hemiperfect
3722:Tesseractic
3660:Icosahedral
3640:Tetrahedral
3571:Dodecagonal
3272:Recursively
3143:Prime power
3118:Sixth power
3113:Fifth power
3093:Power of 10
3051:Classes of
2856:pp. 773-774
2646:carry digit
2512:Synergetics
1788:. A number
1485:Any number
1430:other than
1422:Other bases
37:palindromic
4959:Categories
4910:Graphemics
4783:Pernicious
4637:Undulating
4612:Pandigital
4586:Trimorphic
4187:Nontotient
4036:Arithmetic
3650:Octahedral
3551:Heptagonal
3541:Pentagonal
3526:Triangular
3367:Sierpiński
3289:Jacobsthal
3088:Power of 3
3083:Power of 2
2933:References
2909:2021-04-28
2898:1602.06208
2811:2008.06864
2704:(sequence
2684:PrimePages
2650:spillovers
1267:for which
628:squarefree
124:there are
41:palindrome
4667:Parasitic
4516:Factorion
4443:Digit sum
4435:Digit sum
4253:Fortunate
4240:Primorial
4154:Semiprime
4091:Practical
4056:Descartes
4051:Deficient
4041:Betrothed
3883:Wieferich
3712:Pentatope
3675:pyramidal
3566:Decagonal
3561:Nonagonal
3556:Octagonal
3546:Hexagonal
3405:Practical
3352:Congruent
3284:Fibonacci
3248:Loeschian
2975:MathWorld
2950:, S. 61 (
2521:#, where
2516:primorial
2432:−
2421:−
2415:−
2322:−
2140:−
2077:−
2041:≤
2030:for some
1941:−
1935:≤
1929:≤
1859:−
1853:≤
1847:≤
1633:−
1597:(because
1582:−
1418:> 1).
201:∑
4739:Friedman
4672:Primeval
4617:Repdigit
4574:-related
4521:Kaprekar
4495:Meertens
4418:Additive
4405:dynamics
4313:Friendly
4225:Sociable
4215:Amicable
4026:Abundant
4006:dynamics
3828:Schröder
3818:Narayana
3788:Eulerian
3778:Delannoy
3773:Dedekind
3594:centered
3460:centered
3347:Amenable
3304:Narayana
3294:Leonardo
3190:Mersenne
3138:Powerful
3078:Achilles
2919:Archived
2903:Archived
2859:Archived
2854:Fuller,
2831:Archived
2368:in base
1617:is then
1478:and the
1323:, where
305:for all
266:for all
130:sequence
4912:related
4876:related
4840:related
4838:Sorting
4723:Vampire
4708:Harshad
4650:related
4622:Repunit
4536:Lychrel
4511:Dudeney
4363:Størmer
4358:Sphenic
4343:Regular
4281:divisor
4220:Perfect
4116:Sublime
4086:Perfect
3813:Motzkin
3768:Catalan
3309:Padovan
3243:Leyland
3238:Idoneal
3233:Hilbert
3205:Woodall
2791:2325149
2744:(ed.).
2710:in the
2707:A065379
2690:11 July
2491:in the
2488:A118031
2475:A281508
2471:A281509
2207:) are:
2203:in the
2200:A016038
1466:in the
1463:A006995
1448:in the
1445:A057148
1432:decimal
1393:in the
1390:A002477
1379:in the
1376:A186080
1362:in the
1359:A002781
1345:in the
1342:A002779
1337:squares
538:111110
458:199999
455:109999
427:natural
388:
378:in the
375:A070199
323:base 10
300:−
152:in any
142:decimal
112:in the
109:A002779
95:in the
92:A002385
65:in the
62:A002113
55:) are:
53:decimal
49:racecar
4778:Odious
4703:Frugal
4657:Cyclic
4646:Digit-
4353:Smooth
4338:Pronic
4298:Cyclic
4275:Other
4248:Euclid
3898:Wilson
3872:Primes
3531:Square
3400:Polite
3362:Riesel
3357:Knödel
3319:Perrin
3200:Thabit
3185:Fermat
3175:Cullen
3098:Square
3066:Powers
2946:
2841:
2789:
2729:p. 520
1436:binary
780:)=-1)
653:12160
547:square
535:61110
532:11110
498:88889
495:48889
452:19999
449:10999
173:digits
4819:Prime
4814:Lucky
4803:sieve
4732:Other
4718:Smith
4598:Digit
4556:Happy
4531:Keith
4504:Other
4348:Rough
4318:Giuga
3783:Euler
3645:Cubic
3299:Lucas
3195:Proth
2893:arXiv
2806:arXiv
2787:JSTOR
2665:Notes
1525:with
1354:cubes
1296:5683
1293:1417
1213:3292
1210:1762
1175:2814
1172:2001
1137:1400
1134:1056
1070:2403
1067:1768
1032:4221
1029:2392
994:1782
956:5607
953:3070
918:4315
915:2428
880:4452
877:2486
842:6067
839:1010
804:6067
801:3438
762:6093
759:3383
738:)=1)
699:7839
696:4178
650:6821
647:1200
619:5953
600:prime
529:6110
526:1110
492:8889
489:4889
446:1999
443:1099
45:rotor
27:or a
4773:Evil
4453:Self
4403:and
4293:Blum
4004:and
3808:Lobb
3763:Cake
3758:Bell
3508:Star
3415:Ulam
3314:Pell
3103:Cube
2944:ISBN
2839:ISBN
2750:The
2712:OEIS
2692:2023
2577:1001
2493:OEIS
2205:OEIS
2176:1001
2047:<
1986:>
1751:1991
1659:1221
1536:>
1474:The
1468:OEIS
1450:OEIS
1410:for
1395:OEIS
1381:OEIS
1364:OEIS
1347:OEIS
1290:688
1287:114
1207:348
1204:173
1169:390
1166:250
1131:179
1128:122
1099:413
1064:303
1061:205
1026:412
1023:226
991:991
988:171
950:566
947:317
912:437
909:251
874:482
871:268
836:180
833:100
798:617
795:351
756:583
753:324
693:799
690:424
675:=0)
644:675
641:120
616:781
613:113
575:cube
523:610
520:110
486:889
483:489
467:even
440:199
437:109
380:OEIS
311:Zero
270:and
183:as:
162:base
122:base
114:OEIS
97:OEIS
85:The
67:OEIS
4891:Ban
4279:or
3798:Lah
2779:doi
2629:001
2623:006
2617:015
2611:020
2605:015
2599:006
2495:).
2390:iff
2300:293
2296:283
2292:269
2288:263
2284:223
2280:179
2276:167
2272:163
2268:149
2264:139
2260:137
2256:103
2194:).
1914:if
1828:if
1728:104
1672:151
1284:47
1281:10
1201:34
1198:12
1163:44
1160:18
1125:24
1122:14
1096:64
1093:11
1058:39
1055:25
1020:41
1017:24
985:98
982:15
979:11
944:56
941:28
906:43
903:23
868:49
865:21
830:21
792:64
789:32
750:56
747:35
734:(μ(
687:79
684:42
638:67
635:12
610:20
566:31
563:20
560:15
557:14
517:60
514:10
507:odd
480:89
477:49
434:19
431:10
418:10
415:10
412:10
409:10
406:10
403:10
400:10
397:10
394:10
391:10
171:+1
79:and
47:or
4961::
3013:,
2972:.
2901:.
2891:.
2887:.
2785:.
2775:96
2773:.
2748:.
2727:,
2681:.
2477:.
2466:.
2449:.
2298:,
2294:,
2290:,
2286:,
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2278:,
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2270:,
2266:,
2262:,
2258:,
2254:,
2252:79
2250:,
2248:53
2246:,
2244:47
2242:,
2240:19
2238:,
2236:11
2234:,
2230:,
2226:,
2222:,
2218:,
2214:,
1956:.
1774:16
1755:10
1742:,
1724:11
1715:34
1711:33
1702:20
1698:55
1689:14
1685:77
1626:11
1397:)
1305:+
1302:+
1299:+
1278:6
1269:σ(
1259:1
1256:1
1253:1
1250:1
1247:1
1244:0
1241:0
1238:0
1235:0
1232:0
1219:+
1216:+
1195:1
1192:0
1181:+
1178:+
1157:1
1154:0
1143:+
1140:+
1119:3
1116:1
1105:+
1102:+
1090:3
1087:2
1076:+
1073:+
1052:4
1049:1
1038:+
1035:+
1014:4
1011:1
1000:+
997:+
976:2
973:1
962:+
959:+
938:5
935:4
924:+
921:+
900:4
897:3
886:+
883:+
862:4
859:3
848:+
845:+
827:9
824:2
821:1
810:+
807:+
786:6
783:4
768:+
765:+
744:6
741:2
722:5
719:3
716:2
705:+
702:+
681:7
678:4
669:μ(
659:+
656:+
632:6
607:5
604:4
591:8
588:7
585:5
582:4
579:3
554:7
551:4
511:5
474:9
471:5
345:.
343:11
309:.
116:).
99:).
69:).
19:A
3163:a
3044:e
3037:t
3030:v
2978:.
2954:)
2925:)
2914:(
2912:.
2895::
2845:.
2814:.
2808::
2793:.
2781::
2731:.
2694:.
2626:,
2620:,
2614:,
2608:,
2602:,
2596:,
2593:1
2590:=
2585:6
2581:)
2574:(
2545:n
2523:n
2519:n
2435:i
2429:k
2425:a
2418:1
2412:b
2409:=
2404:i
2400:a
2376:b
2354:i
2350:a
2325:1
2319:b
2232:6
2228:4
2224:3
2220:2
2216:1
2212:0
2180:2
2172:=
2169:9
2166:=
2163:n
2143:1
2137:a
2127:n
2111:2
2107:a
2103:=
2100:n
2080:1
2074:b
2064:n
2050:b
2044:a
2038:2
2018:b
2015:a
2012:=
2009:n
1989:6
1983:n
1969:n
1965:b
1961:b
1944:2
1938:p
1932:b
1924:p
1902:b
1882:p
1862:2
1856:n
1850:b
1844:2
1840:/
1836:n
1816:b
1796:n
1770:7
1766:C
1763:7
1760:=
1720:=
1707:=
1694:=
1681:=
1676:8
1668:=
1663:4
1636:1
1630:n
1605:n
1585:1
1579:n
1559:n
1539:n
1533:b
1513:b
1493:n
1470:)
1452:)
1416:n
1412:k
1408:n
1383:)
1366:)
1349:)
1329:k
1325:n
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1273:)
1271:n
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1187:n
1149:n
1111:n
1082:n
1044:n
1006:n
968:n
930:n
892:n
854:n
816:n
778:n
774:n
736:n
728:n
711:n
673:)
671:n
665:n
625:n
597:n
572:n
544:n
504:n
464:n
424:n
307:i
302:i
298:k
294:a
289:i
285:a
281:n
276:k
272:a
268:i
264:b
259:i
255:a
236:i
232:b
226:i
222:a
216:k
211:0
208:=
205:i
197:=
194:n
180:i
176:a
169:k
165:b
158:n
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