A127106 -id:A127106 - 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

A127837 Numbers k such that ((k+1)^k-1)/k^2 is a prime.

Original entry on oeis.org

2, 3, 5, 17, 4357

Offset: 1

Farideh Firoozbakht and Alexander Adamchuk, Mar 13 2007

All terms are primes. Corresponding primes of the form ((k+1)^k-1)/k^2 are listed in A128466 = 2, 7, 311, 7563707819165039903, ... .
It seems that if p is in the sequence then the first three numbers k such that k^2 divides (p+1)^k-1 are: 1, p & ((p+1)^p-1)/p. 2 is in the sequence and the first three terms of A127103 are : 1, 2 & ((2+1)^2-1)/2; 3 is in the sequence and the first three terms of A127104 are : 1, 3 & ((3+1)^3-1)/3; 5 is in the sequence and the first three terms of A127106 are : 1, 5 & ((5+1)^5-1)/5.
No other terms below 20000. - Max Alekseyev, Apr 25 2007

			4357 is in the sequence because (4358^4357-1)/4357^2 is prime.

Cf. A128466, A037205, A060072, A060073, A058128, A128456, A127103, A127104, A127106, A128398.

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.

A292332 Numbers k such that k^2 divides 6^k + 1.

Original entry on oeis.org

1, 7, 203, 1379, 11977, 39991, 2359469, 4780447, 6521291, 6640739, 9904979, 28434413, 189117439, 282046373, 391803601, 941748059, 1308225583, 1677630367, 1951280863, 3524747261, 5601579361, 7033936399, 11157928901, 11580995069, 55563135481, 77185309397, 114689751841, 156382762711, 233252350471, 324021578027, 330493182299, 335848857001, 669601991737, 694375210417, 930174952469

Offset: 1

Max Alekseyev, Sep 14 2017

nonn

Max Alekseyev, Table of n, a(n) for n = 1..46

Cf. A062394. Subsequence of A015953.

Cf. A127106, A128680.

A333502 a(n) is the n-th number m such that m^2 divides n^m - 1 (or 0 if m does not exist).

Original entry on oeis.org

1, 0, 4, 903, 12, 776119592182705, 12, 42931441, 136, 27486820443, 60, 107342336783, 84

Offset: 1

Seiichi Manyama, Mar 24 2020

Main diagonal of A333500.

Cf. A127106, A127100, A128405, A333433.

{a(n) = if(n==2, 0, my(cnt=0, k=0); while(cnt

a(n) = A333500(n,n).

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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A127837 Numbers k such that ((k+1)^k-1)/k^2 is a prime.

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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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A292332 Numbers k such that k^2 divides 6^k + 1.

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A333502 a(n) is the n-th number m such that m^2 divides n^m - 1 (or 0 if m does not exist).

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