A128400 -id:A128400 - cp's OEIS frontend

A128395 Numbers k such that k^2 divides 15^k-1.

1, 2, 4, 7, 8, 14, 16, 28, 56, 112, 136, 272, 452, 812, 904, 952, 1624, 1808, 1904, 3164, 3248, 6328, 11912, 12656, 15368, 18632, 23824, 27608, 30736, 37264, 47908, 55216, 60248, 83384, 91756, 95816, 102604, 107576, 113936, 120496, 130424, 166768

Offset: 1

Alexander Adamchuk, Mar 01 2007

Amiram Eldar, Table of n, a(n) for n = 1..640

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

a={1};For[n=1,n<200000,n++,If[PowerMod[15,n,n^2]==1, AppendTo[a, n]]]; a (* Stefan Steinerberger, Jun 10 2007 *)
Join[{1},Select[Range[167000],PowerMod[15,#,#^2]==1&]] (* Harvey P. Dale, Sep 14 2020 *)

is(k) = Mod(15, k^2)^k == 1; \\ Amiram Eldar, May 21 2024

More terms from Stefan Steinerberger, Jun 10 2007

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

Original entry on oeis.org

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

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

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.

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A128395 Numbers k such that k^2 divides 15^k-1.

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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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A128360 Numbers k such that k divides 20^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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