A128395 -id:A128395 - cp's OEIS frontend

A128404 Numbers k such that k^2 divides 24^k-1.

1, 23, 1081, 2870377, 7009273, 15954479, 134907719, 329435831, 537539141, 15001987199, 874750261127, 1991103024721, 4272172921319, 4862143429729, 7933540182019, 12816504745411, 41113262272969, 67084347257659

Offset: 1

Alexander Adamchuk, Mar 01 2007

23 divides all terms except the first.

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

a={}; Do[r=(24^n-1)/n^2; If[r==IntegerPart[r], AppendTo[a, n]], {n, 1, 10^3}]; a (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)

a(5)-a(6) from Farideh Firoozbakht, Mar 05 2007
a(7)-a(10) from Ryan Propper, Feb 23 2008
Terms a(11) onward from Max Alekseyev, May 06 2010

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.

cp's OEIS Frontend

A128404 Numbers k such that k^2 divides 24^k-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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