A127102 -id:A127102 - cp's OEIS frontend

A177913 Numbers k such that k^3 divides 13^(k^2) - 1.

1, 2, 3, 4, 6, 10, 12, 14, 20, 28, 30, 34, 42, 60, 68, 70, 84, 102, 110, 114, 140, 170, 183, 204, 210, 220, 222, 228, 238, 330, 340, 366, 406, 420, 444, 476, 510, 570, 660, 714, 732, 770, 798, 812, 820, 876, 930, 942, 1010, 1020, 1110, 1140, 1190, 1218, 1428

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

Robert Price, Table of n, a(n) for n = 1..2236

Cf. A128393 (k^2 divides 13^k-1).
Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.
Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

Join[{1}, Select[Range[3000000], PowerMod[13, #^2, #^3] == 1 &]] (* Robert Price, Mar 31 2020 *)

A177915 Numbers k such that k^3 divides 15^(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 28, 56, 68, 112, 136, 226, 272, 406, 452, 476, 812, 904, 952, 1582, 1624, 1808, 1904, 2056, 2758, 3164, 3248, 4112, 5516, 5956, 6328, 7684, 9316, 11032, 11912, 12656, 13804, 14392, 15368, 18632, 21512, 22064, 23824, 23954, 25144

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

Robert Price, Table of n, a(n) for n = 1..205

Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.

Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

Join[{1}, Select[Range[30000], PowerMod[15, #^2, #^3] == 1 &]] (* Robert Price, Apr 04 2020 *)

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

Alexander Adamchuk, May 14 2010

17 divides a(n) for n > 1.

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.

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

Three more terms from Max Alekseyev, Oct 02 2010

A177919 Numbers k such that k^3 divides 19^(k^2) - 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 10, 12, 18, 20, 30, 36, 42, 60, 68, 78, 84, 90, 110, 126, 156, 180, 204, 210, 220, 222, 234, 252, 294, 330, 340, 362, 381, 390, 420, 438, 444, 468, 546, 588, 612, 630, 654, 660, 666, 724, 762, 780, 820, 876, 882, 930, 990, 1010, 1014, 1020

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

Robert Price, Table of n, a(n) for n = 1..4378

Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.

Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

Select[ Range[10^4], Mod[ PowerMod[ 19, #^2, #^3 ] - 1, #^3 ] == 0 &]

A128456 Quotients A128452(p+1)/p for prime p = A000040(n).

Original entry on oeis.org

2, 7, 311, 127, 23, 157, 7563707819165039903, 75368484119, 47, 9629, 311, 25679, 821, 758771382833029, 12409, 71233, 18438666190697, 2443783, 2939291, 71711, 352883, 181113265579, 167, 105199, 3881, 1314520253, 619, 20759, 117503, 1162660843

Offset: 1

Alexander Adamchuk, Mar 05 2007, Mar 09 2007

nonn

a(n) coincides with A128357(n) from n = 2 up to n = 6.

Cf. A128452, A128357, A128356, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

a(n) = A128452(A000040(n)+1)/A000040(n).

a(n) = A020639(((p+1)^p - 1)/p^2), i.e., the smallest prime factor of ((p+1)^p - 1)/p^2, where p = A000040(n).

Terms a(14) onward from Max Alekseyev, May 05 2010

A128452 Least number k > n such that k^2 divides n^k - 1.

Original entry on oeis.org

4, 21, 6, 1555, 8, 889, 10, 111, 12, 253, 14, 2041, 16, 21, 18, 128583032925805678351, 20, 1432001198261, 22, 39, 24, 1081, 26, 55, 28, 171, 30, 279241, 32, 9641, 34, 1191, 36, 55, 38, 950123, 40, 1641, 42, 33661, 44, 32627169461820247, 46, 63, 48, 583223, 50

Offset: 3

Alexander Adamchuk, Mar 05 2007, Mar 09 2007

nonn

For prime p, p divides a(p+1). Quotients a(p+1)/p for prime p = A000040(n) are listed in A128456(n) which coincides with A128357(n) for n from 2 to 6.

a(n) divides n^(n-1) - 1.

Cf. A128456, A128357, A128356, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

a(2n-1) = 2n.

More terms from Alexander Adamchuk, Mar 09 2007

Terms a(22) onward from Max Alekseyev, May 05 2010

A333500 A(n,k) is the n-th number m such that m^2 divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 0, 3, 1, 2, 0, 4, 1, 3, 4, 0, 5, 1, 2, 21, 20, 0, 6, 1, 5, 4, 903, 220, 0, 7, 1, 2, 1555, 6, 2667, 1220, 0, 8, 1, 7, 3, 9673655, 12, 7077, 2420, 0, 9, 1, 2, 889, 4, 187159211791705, 42, 113799, 5060, 0, 10, 1, 3, 4, 2359, 6, 776119592182705, 52, 114681, 13420, 0, 11

Offset: 1

Seiichi Manyama, Mar 24 2020

			Square array A(n,k) begins:
  1, 1,    1,    1,  1,               1, ...
  2, 0,    2,    3,  2,               5, ...
  3, 0,    4,   21,  4,            1555, ...
  4, 0,   20,  903,  6,         9673655, ...
  5, 0,  220, 2667, 12, 187159211791705, ...
  6, 0, 1220, 7077, 42, 776119592182705, ...

Columns k=1-24 give: A000027, A063524, A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.
Main diagonal gives A333502.
Cf. A333432.

A177908 Integers n such that n^3 divides 8^(n^2) - 1.

Original entry on oeis.org

1, 7, 889, 2359, 299593, 2033143, 13549249, 42931441, 100170217, 188097287, 233727361, 310935751, 685169191, 1515836567, 3606045247, 4566096913, 5452293007, 6620620783, 12721617559, 13162910047, 24088984969, 29683374847, 30987132463, 63388785719, 65576560063, 92349997537

Offset: 1

Max Alekseyev, May 17 2010

nonn

Contains A127102 as a subsequence.
From M. F. Hasler, Nov 21 2018: (Start)
The first terms not in A127102 are a({10, 11, 14, 20, 21, 22, ...}) = {188097287, 233727361, 1515836567, 13162910047, 24088984969, 29683374847, ...}.
The listed terms are all squarefree, and all but the first two terms appear to be divisible by either a(3) = 7*127 or a(4) = 7*337. Are there exceptions to these properties? (End)

Cf. A129211, A129212, A177905, A177907, A177909, A177243, A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

Select[Range[2 10^5], IntegerQ[(8^(#^2) - 1) / #^3] &] (* or *) Select[Range[2 10^6], IntegerQ[(PowerMod[8, #, #^2] - 1) / #^3] &] (* Vincenzo Librandi, Nov 23 2018 *)

is(n)=Mod(8,n^3)^n^2==1 \\ M. F. Hasler, Nov 21 2018

a(23)-a(26) from Giovanni Resta, Nov 23 2018

A177164 a(n) = (n^r - 1)/r^2, where r = (n^(n-1) - 1)/(n-1).

Original entry on oeis.org

1, 5, 9972894583, 449853889404077636694265177903207995382439448590987815041588427345865911961016023550064137351211162870609

Offset: 2

Alexander Adamchuk, May 04 2010

nonn

The next term has 1204 digits.
r = (n^(n-1) - 1)/(n-1) = A060072(n) is the (n-1)-digit repunit in base n.
r^2 divides n^r - 1 for all bases n > 1.

			a(10) = (10^111111111 - 1)/111111111^2.

Cf. A060072, A127100, A127101, A127102, A127103, A127104, A127105, A127106, A127107, A127092.

Table[(n^((n^(n - 1) - 1)/(n - 1)) - 1)/((n^(n - 1) - 1)/(n - 1))^2, {n, 2, 6}]

a(n) = (n^((n^(n-1) - 1)/(n-1)) - 1)/((n^(n-1) - 1)/(n-1))^2.

a(n) = (n^A060072(n) - 1)/A060072(n)^2.

A177906 Numbers k such that k^3 divides 6^(k^2) - 1.

Original entry on oeis.org

1, 5, 1555, 9673655, 24181805, 90993505, 200928005, 28298980055, 36850702555, 62488609555, 141377087255, 150435008905, 367279622065, 2256331679135, 7521049172905, 18802586659555, 24599612913355, 54706615318945

Offset: 1

Alexander Adamchuk, May 16 2010

nonn

5 divides a(n) for n > 1.
Prime divisors of a(n) in order of their appearance are {5, 311, 6221, 15551, 18198701, 40185601, ...}.
From there on, the list is no longer increasing; it continues with 23698201, 90917741, 236192683, 93307, 311021, ... - M. F. Hasler, Oct 21 2012

Cf. A014946 (k divides 6^k-1), A127106 (k^2 divides 6^k-1).
Cf. A129211, A129212, A177905, A177907, A127102, A177909, A177243.
Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

is_A177906(n)=Mod(6,n^3)^(n^2)==1  \\ M. F. Hasler, Oct 21 2012

More terms from Max Alekseyev, Oct 02 2010

cp's OEIS Frontend

A177913 Numbers k such that k^3 divides 13^(k^2) - 1.

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A177915 Numbers k such that k^3 divides 15^(k^2) - 1.

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

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A177919 Numbers k such that k^3 divides 19^(k^2) - 1.

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A128456 Quotients A128452(p+1)/p for prime p = A000040(n).

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A128452 Least number k > n such that k^2 divides n^k - 1.

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A333500 A(n,k) is the n-th number m such that m^2 divides k^m - 1 (or 0 if m does not exist); square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A177908 Integers n such that n^3 divides 8^(n^2) - 1.

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A177164 a(n) = (n^r - 1)/r^2, where r = (n^(n-1) - 1)/(n-1).

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A177906 Numbers k such that k^3 divides 6^(k^2) - 1.

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