A014949 -id:A014949 - cp's OEIS frontend

A127102 Numbers k such that k^2 divides 8^k-1.

1, 7, 889, 2359, 299593, 2033143, 13549249, 42931441, 100170217, 310935751, 685169191, 3606045247, 4566096913, 5452293007, 6620620783, 12721617559, 30987132463, 65576560063, 92349997537, 104785348087, 457967746369

Offset: 1

Alexander Adamchuk, Jan 05 2007, Jan 07 2007

nonn

Subsequence of A014949.
a(5) = 299593 = (8^7-1)/7 is a repunit in base 8, 1111111(base 8). The first 5 listed terms of a(n) are terms of the finite sequence A003530: divisors of 8^7-1. [Corrected and edited by M. F. Hasler, Nov 21 2018]
Also a subsequence of A177908, see there for more comments on properties of terms of this sequence. - M. F. Hasler, Nov 21 2018

Cf. A127100, A127101, A127103, A127104, A127105, A127106, A127107, A127092, A014949, A003530.

Select[Range[20000], IntegerQ[(PowerMod[8, #, #^2 ]-1)/#^2 ]&]

is(n)=Mod(8,n^2)^n==1 \\ M. F. Hasler, Nov 21 2018

More terms from Ryan Propper, Jan 05 2007

Terms a(12) onward from Max Alekseyev, May 06 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

Alexander Adamchuk, Mar 02 2007

nonn

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

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

is(n)=Mod(20,n)^n==1 \\ Charles R Greathouse IV, Nov 04 2016

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

Olivier Gérard

nonn

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

Max Alekseyev, Table of n, a(n) for n = 1..146

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).

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 *)

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

Olivier Gérard

nonn

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

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

Cf. A177914, A128394.

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

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

A068382 Numbers k such that k divides 9^k - 1.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 80, 100, 110, 128, 136, 160, 164, 200, 220, 250, 256, 272, 320, 328, 400, 440, 500, 512, 544, 550, 610, 640, 656, 680, 800, 820, 880, 1000, 1024, 1088, 1100, 1210, 1220, 1250, 1280, 1312, 1360, 1544, 1600, 1640, 1760

Offset: 1

Benoit Cloitre, Mar 05 2002

For all m the sequence includes 2^m, 10^m, 2*10^m, 10*2^m.

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A014945, A014946, A014949, A014950.

Join[{1}, Select[Range[10000], PowerMod[9, #, #] == 1 &]] (* Robert Price, Apr 04 2020 *)

isok(n) = Mod(9, n)^n == Mod(1, n); \\ Michel Marcus, May 06 2016

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

Olivier Gérard

nonn

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

Harvey P. Dale, Table of n, a(n) for n = 1..2000

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

Cf. A128396, A177916

Join[{1},Select[Range[1000],PowerMod[16,#,#]==1&]] (* Harvey P. Dale, Jun 12 2024 *)

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

Edited by Max Alekseyev, Sep 10 2011

A068383 Numbers k such that k divides 11^k - 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 16, 18, 20, 24, 25, 30, 32, 36, 40, 42, 48, 50, 54, 60, 64, 72, 80, 84, 90, 96, 100, 108, 114, 120, 125, 126, 128, 144, 150, 156, 160, 162, 168, 180, 192, 200, 210, 216, 222, 228, 240, 244, 250, 252, 256, 270, 272, 288, 294, 300, 312, 320

Offset: 1

Benoit Cloitre, Mar 05 2002

For all k, 2^k, 10^k, 2 * 3^k and 10 * 3^k are in the sequence.

			11^5 - 1 = 161050, which is divisible by 5, so 5 is in the sequence.
11^6 - 1 = 1771560, which is divisible by 6, so 6 is in the sequence.
11^7 = 19487171 = 4 modulo 7, so 7 is not in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A014945, A014946, A014949, A014950, A068382.

Join[{1}, Select[Range[500], PowerMod[11, #, #] == 1 &]] (* Robert Price, Apr 04 2020 *)

isok(n) = Mod(11, n)^n == Mod(1, n); \\ Michel Marcus, May 06 2016

def powerMod(a: Int, b: Int, m: Int): Int = b match { case 1 => a % m; case n => a * powerMod(a, n - 1, m) % m }
List(1) ++: (2 to 500).filter(k => powerMod(11, k, k) == 1) // Alonso del Arte, Apr 04 2020

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

Alexander Adamchuk, Mar 02 2007

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.

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

Cf. A128357 (quotients A128356(n)/prime(n)).

(* 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 *)

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

Alexander Adamchuk, Mar 02 2007, Mar 09 2007

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).

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)).

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

Alexander Adamchuk, May 17 2010

nonn

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

Robert Israel, Table of n, a(n) for n = 1..2500

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

Contains A003591.

filter:= n -> 15 &^ n - 1 mod n = 0:
select(filter, [$1..10000]); # Robert Israel, Feb 25 2020

is(n)=Mod(15,n)^n==1 \\ Charles R Greathouse IV, Nov 04 2016

cp's OEIS Frontend

A127102 Numbers k such that k^2 divides 8^k-1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A068382 Numbers k such that k divides 9^k - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A068383 Numbers k such that k divides 11^k - 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Scala

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).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica