A128405 -id:A128405 - cp's OEIS frontend

A128402 Numbers k such that k^2 divides 22^k-1.

1, 3, 7, 21, 39, 273, 507, 3081, 3549, 21567, 40053, 78117, 280371, 343239, 546819, 1015521, 2056899, 2402673, 5998317, 6171243, 7108647, 8740173, 12338859, 14398293, 18988203, 27115881, 41988219, 43198701, 47727771, 55431363

Offset: 1

Alexander Adamchuk, Mar 01 2007

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

select(t -> 22 &^ t - 1 mod t^2 = 0, [seq(2*k+1,k=0..10^6)]); # Robert Israel, Jan 23 2015

a={}; Do[r=(22^n-1)/n^2; If[r==IntegerPart[r], AppendTo[a, n]], {n, 1, 10^3}]; a (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)

{ forstep(m=11,10^8,2, if( Mod(22,m^2)^m==1, print(m) ) ) } \\ Max Alekseyev, Oct 18 2008

a(14)-a(30) from Max Alekseyev, Oct 18 2008

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

Original entry on oeis.org

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

A177912 Numbers k such that k^3 divides 12^(k^2) - 1.

Original entry on oeis.org

1, 11, 253, 11891, 77209, 768361, 1775807, 17666737, 36112967, 61488361, 83462929, 154261943, 173185331, 591303757, 830336639, 971656873, 2936791979, 4139054953, 5393125859, 8139710557, 8142849781, 45667873031, 53653880269

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

11 divides a(n) for n > 1.

Cf. A014951 (k divides 12^k-1), A128405 (k^2 divides 12^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[12, #^2, #^3] == 1 &]] (* Robert Price, Mar 31 2020 *)

More terms from Max Alekseyev, Oct 02 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.

A292337 Numbers k such that k^2 divides 12^k + 1.

Original entry on oeis.org

1, 13, 1027, 468481, 2890927, 37009999, 228383233, 2126334691, 104180336299, 8230246567621, 34238014256797, 76626723259567, 131147819778859, 219609509027857, 472852182249889, 1033753383066391, 2704803126286963, 10360677762529861, 16436880348770011, 17349151213200703, 81666517262244889, 686779630199762359

Offset: 1

Max Alekseyev, Sep 16 2017

nonn

Max Alekseyev, Table of n, a(n) for n = 1..29

Subsequence of A015961.

Cf. A128405, A128685.

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).

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A128402 Numbers k such that k^2 divides 22^k-1.

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

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A177912 Numbers k such that k^3 divides 12^(k^2) - 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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A292337 Numbers k such that k^2 divides 12^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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