cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

A134360 Primes dividing terms of A128358.

Original entry on oeis.org

17, 20231, 83233, 3828719, 8519143, 39951841, 68916943, 76672279, 144326023, 721630111, 35915273857, 39392517233, 42845049953, 83709099119
Offset: 1

Views

Author

Alexander Adamchuk, Jan 31 2008

Keywords

Comments

A prime p is in this sequence if the multiplicative order of 18 modulo p equals the product of smaller primes from this sequence. - Max Alekseyev, Sep 24 2009

Examples

			A128358 begins with {1, 17, 17^2, 17^3, 17^4, 17^5, 20231*17^2, 83233*17^2, ...}. Thus 17, 20231, 83233 are in this sequence.

Crossrefs

Cf. A128356, A128357, A128358.

Extensions

Edited and extended by Max Alekseyev, Jul 28 2009
a(11)-a(14) from Max Alekseyev, May 26 2010

A128398 Numbers k such that k^2 divides 18^k - 1.

Original entry on oeis.org

1, 17, 128583032925805678351
Offset: 1

Views

Author

Alexander Adamchuk, Mar 08 2007

Keywords

Comments

17 divides a(n) for n > 1.
a(4) > 10^50. - Max Alekseyev, Mar 01 2025

Crossrefs

Subsequence of A128358.
Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

A177918 Numbers k such that k^3 divides 18^(k^2) - 1.

Original entry on oeis.org

1, 17, 343927, 1414961, 28626075991, 610559655569, 5417488064959
Offset: 1

Views

Author

Alexander Adamchuk, May 14 2010

Keywords

Comments

17 divides a(n) for n > 1.

Crossrefs

Cf. A128358 (k divides 18^k - 1), A128398 (k^2 divides 18^k - 1).
Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.
Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

Programs

  • Mathematica
    Select[Range[350000], Mod[PowerMod[18, #^2, #^3] - 1, #^3] == 0 &] (* Julien Kluge, Sep 20 2016 *)

Extensions

Three more terms from Max Alekseyev, Oct 02 2010

A128360 Numbers k such that k divides 20^k - 1.

Original entry on oeis.org

1, 19, 361, 6859, 130321, 2476099, 47045881, 148305659, 893871739, 2817807521, 4234136149, 10350100679, 16983563041, 53538342899, 80448586831, 196651912901, 322687697779, 815211156289, 1017228515081, 1432001198261, 1528523149789
Offset: 1

Views

Author

Alexander Adamchuk, Mar 02 2007

Keywords

Comments

19 divides a(n) for n > 1. All powers of 19 are terms. a(n) = 19^(n-1) for all to n < 8, while a(8) = A128356(8) = 148305659 = 410819*19^2.
Prime divisors of a(n) in the order of appearance are {19, 410819, 617311, 1508981, ...}. - Alexander Adamchuk, May 16 2010

Crossrefs

Cf. A001029, A128358, A014960, A014956, A014951, A014949, A014946, A014945, A067945, A128357, A128356, A128400, A177920.

Programs

Extensions

a(9)-a(11) from Stefan Steinerberger, May 09 2007
a(12)-a(15) from Alexander Adamchuk, May 16 2010
Edited and a(16)-a(21) added by Max Alekseyev, Oct 02 2010

A014960 Integers n such that n divides 24^n - 1.

Original entry on oeis.org

1, 23, 529, 1081, 12167, 24863, 50807, 279841, 571849, 1168561, 2387929, 2870377, 6436343, 7009273, 13152527, 15954479, 26876903, 54922367, 66018671, 112232663, 134907719, 148035889, 161213279, 302508121, 329435831
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Also, numbers n such that n divides s(n), where s(1)=1, s(k)=s(k-1)+k*24^(k-1) (cf. A014942).
All n > 1 in the sequence are multiple of 23. - Conjectured by Thomas Baruchel, Oct 10 2003; proved by Max Alekseyev, Nov 16 2019
If n is a term and prime p|(24^n - 1), then n*p is a term. In particular, if n is a term and prime p|n, then n*p is a term. The smallest term with 3 distinct prime factors is a(16) = 15954479 = 23 * 47 * 14759. - Max Alekseyev, Nov 16 2019

Links

Crossrefs

Prime factors are listed in A087807.
Cf. A014942.
Integers n such that n divides b^n - 1: A067945 (b=3), A014945 (b=4), A067946 (b=5), A014946 (b=6), A067947 (b=7), A014949 (b=8), A068382 (b=9), A014950 (b=10), A068383 (b=11), A014951 (b=12), A116621 (b=13), A014956 (b=14), A177805 (b=15), A014957 (b=16), A177807 (b=17), A128358 (b=18), A125000 (b=19), A128360 (b=20), A014959 (b=22).

Programs

  • Mathematica
    s = 1; Do[ If[ Mod[ s, n ] == 0, Print[n]]; s = s + (n + 1)*24^n, {n, 1, 100000}]
Join[{1},Select[Range[330*10^6],PowerMod[24,#,#]==1&]] (* Harvey P. Dale, Jan 19 2023 *)

Extensions

More terms from Robert G. Wilson v, Sep 13 2000
a(9)-a(12) from Thomas Baruchel, Oct 10 2003
Edited and terms a(13) onward added by Max Alekseyev, Nov 16 2019

A014956 Positive integers k such that k divides 14^k - 1.

Original entry on oeis.org

1, 13, 169, 2041, 2197, 26533, 28561, 114413, 320437, 344929, 371293, 1487369, 4165681, 4484077, 4826809, 17962841, 19335797, 24355253, 50308609, 54153853, 58293001, 62748517, 77457601, 233516933, 249302027, 251365361, 316618289
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Also, positive integers k such that k divides A014929(k).
13 divides a(n) for n > 1. All powers of 13 are terms. All a(n) that are not powers of 13 are divisible either by 157 or 677 or both. - Alexander Adamchuk, May 14 2010
Prime divisors of a(n) in order of appearance: {13, 157, 677, 11933, 122147, 52807, ...}. - Alexander Adamchuk, May 16 2010

Crossrefs

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A177805, A014957, A177807, A128358, A128360.
Cf. A177914, A128394.

Programs

  • Mathematica
    Join[{1}, Select[Range[2000000], PowerMod[14, #, #] == 1 &]] (* Robert Price, Mar 31 2020 *)

Extensions

2 more terms from R. J. Mathar, Mar 05 2008
a(8)-a(23) from Alexander Adamchuk, May 14 2010
a(24)-a(44) from Alexander Adamchuk, May 16 2010
Edited by Max Alekseyev, Sep 10 2011

A014957 Positive integers k that divide 16^k - 1.

Original entry on oeis.org

1, 3, 5, 9, 15, 21, 25, 27, 39, 45, 55, 63, 75, 81, 105, 117, 125, 135, 147, 155, 165, 171, 189, 195, 205, 225, 243, 273, 275, 315, 333, 351, 375, 405, 441, 465, 495, 507, 513, 525, 567, 585, 605, 609, 615, 625, 657, 675, 729, 735, 775, 819, 825, 855, 903
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Also, positive integers k that divide A014931(k).

Links

Crossrefs

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A014956, A177805, A177807, A128358, A128360
Cf. A128396, A177916

Programs

  • Mathematica
    Join[{1},Select[Range[1000],PowerMod[16,#,#]==1&]] (* Harvey P. Dale, Jun 12 2024 *)
  • Python
    A014957_list = [n for n in range(1,10**6) if n == 1 or pow(16,n,n) == 1] # Chai Wah Wu, Mar 25 2021

Extensions

Edited by Max Alekseyev, Sep 10 2011

A128356 Least number k > 1 (that is not the power of prime p) such that k divides (p+1)^k-1, where p = prime(n).

Original entry on oeis.org

20, 21, 1555, 889, 253, 2041, 5846759, 148305659, 1081, 279241, 9641, 950123, 33661, 63213709997, 583223, 3775349, 72707647, 149070763, 196932497, 5091481, 25760459, 14307947980741, 13861, 9362711, 376457, 132766545553, 63757
Offset: 1

Views

Author

Alexander Adamchuk, Mar 02 2007

Keywords

Comments

All listed terms have 2 distinct prime divisors. Most listed terms are semiprimes, except a(7) = 20231*17^2 and a(8) = 410819*19^2. p = prime(n) divides a(n). Quotients a(n)/prime(n) are listed in A128357 = {10, 7, 311, 127, 23, 157, 343927, ...}. a(15) = 583223 = 47*12409. a(16) = 3775349 = 53*71233.

Crossrefs

Cf. A014960, A128360, A128358, A014960, A014956, A014951, A014949, A014946, A014945, A067945.
Cf. A128357 (quotients A128356(n)/prime(n)).

Programs

  • Mathematica
    (* This program is not suitable to compute a large number of terms *) a[n_] := For[p = Prime[n]; k = 2, True, k++, If[Length[FactorInteger[k]] == 2, If[Mod[PowerMod[p + 1, k, k] - 1, k] == 0, Print[k]; Return[k]]]]; Table[a[n], {n, 1, 13}] (* Jean-François Alcover, Oct 07 2013 *)

Extensions

Terms a(14) onwards from Max Alekseyev, Feb 08 2010

A128357 Quotients A128356(n)/prime(n).

Original entry on oeis.org

10, 7, 311, 127, 23, 157, 343927, 7805561, 47, 9629, 311, 25679, 821, 1470086279, 12409, 71233, 1232333, 2443783, 2939291, 71711, 352883, 181113265579, 167, 105199, 3881, 1314520253, 619, 20759, 117503, 1162660843, 1880415721, 263
Offset: 1

Views

Author

Alexander Adamchuk, Mar 02 2007, Mar 09 2007

Keywords

Comments

A128356 = {20, 21, 1555, 889, 253, 2041, 5846759, ...} = Least number k>1 (that is not the power of prime p) such that k divides (p+1)^k-1, where p = prime(n). Most listed terms are primes, except a(7) = 20231*17 and a(8) = 410819*19. a(15) = 12409. a(16) = 71233.
Note that all prime listed terms of {a(n)} coincide with terms of A128456 = {2, 7, 311, 127, 23, 157, 7563707819165039903, 75368484119, 47, 9629, 311, 25679, 821, ...} = least prime factor of ((p+1)^p - 1)/p^2, where p = prime(n).

Crossrefs

Cf. A128356 (least number k > 1 (that is not a power of prime p) such that k divides (p+1)^k-1, where p = prime(n)).
Cf. A014960, A128360, A128358, A014960, A014956, A014951, A014949, A014946, A014945, A067945.
Cf. A128456 (least prime factor of ((p+1)^p - 1)/p^2, where p = prime(n)).

Extensions

Terms a(14) onwards from Max Alekseyev, Feb 08 2010

A177805 Numbers k such that k divides 15^k - 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 32, 49, 56, 64, 98, 112, 128, 136, 196, 224, 256, 272, 343, 392, 448, 452, 512, 544, 686, 784, 812, 896, 904, 952, 1024, 1088, 1372, 1568, 1624, 1792, 1808, 1904, 2048, 2176, 2312, 2401, 2744, 3136, 3164, 3248, 3584, 3616, 3808, 4096
Offset: 1

Views

Author

Alexander Adamchuk, May 17 2010

Keywords

Comments

A000420 are the only odd terms of the sequence. - Robert Israel, Feb 25 2020

Links

Crossrefs

Cf. A000420, A014960, A014956, A014951, A014949, A014946, A014945, A067945, A128358, A128360.
Contains A003591.

Programs

Showing 1-10 of 16 results. Next

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