A127104 -id:A127104 - cp's OEIS frontend

A128394 Numbers k such that k^2 divides 14^k - 1.

1, 13, 2041, 24355253, 249302027, 16772956369, 39665616523, 388885239223, 2974921088191, 3487599163841, 61054982558011, 200151688351277, 473329801968959

Offset: 1

Alexander Adamchuk, Mar 01 2007

13 divides all except the first term.

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

a(4)-a(5) from Farideh Firoozbakht, Mar 05 2007

a(6)-a(13) from Max Alekseyev, May 06 2010

A128396 Numbers k such that k^2 divides 16^k-1.

Original entry on oeis.org

1, 3, 5, 15, 21, 39, 55, 105, 155, 165, 195, 205, 273, 465, 609, 615, 903, 915, 1155, 1265, 1365, 1705, 2067, 2145, 2255, 2265, 2373, 2667, 3045, 3081, 3255, 3795, 4305, 4515, 4895, 4965, 5115, 6045, 6123, 6355, 6405, 6765, 7077, 7455, 7917, 7995, 10065

Offset: 1

Alexander Adamchuk, Mar 01 2007

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

Select[Range[11000],Divisible[16^#-1,#^2]&] (* Harvey P. Dale, Dec 14 2010 *)
Join[{1},Select[Range[11000],PowerMod[16,#,#^2]==1&]] (* Harvey P. Dale, Mar 16 2022 *)

A177917 Numbers k such that k^3 divides 17^(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 40, 42, 48, 58, 60, 72, 78, 80, 84, 90, 110, 114, 116, 120, 126, 144, 156, 168, 174, 180, 210, 220, 222, 228, 232, 234, 240, 252, 290, 312, 330, 336, 342, 348, 360, 390, 420, 440, 444, 456, 464, 468, 504, 522, 546

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

The first 9 terms (and others) of A218087 are in this sequence A177917 as well as in A128397. These two sequences have many terms in common, but A177917 \ A128397 = {10, 30, 58, 90, 110, ...} and A128397 \ A177917 = {580, 624, 660, 684, 696, 720, ...}. - M. F. Hasler, Oct 22 2012

Robert Price, Table of n, a(n) for n = 1..6436

Cf. A129211, A129212, A177905, A177907, A127102, A177909, A177243, A177911, A177912, A177913, A177914, A177915, A177916, A177918, A177919, A177920.

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, ..., A128397.

Select[Range[600],Divisible[17^#^2-1,#^3]&] (* Harvey P. Dale, Jul 19 2015 *)

is_A177917(n)=Mod(17,n^3)^(n^2)==1  \\ M. F. Hasler, Oct 21 2012

A128401 Numbers k such that k^2 divides 21^k-1.

Original entry on oeis.org

1, 2, 4, 5, 10, 20, 22, 44, 52, 68, 110, 220, 260, 340, 506, 572, 748, 820, 884, 1012, 2530, 2756, 2860, 3740, 3916, 4108, 4420, 5060, 9020, 9724, 10660, 13156, 13780, 13940, 17204, 19580, 20540, 23782, 29084, 30316, 34060, 45188, 46852, 47564

Offset: 1

Alexander Adamchuk, Mar 01 2007

Daniel Starodubtsev, Table of n, a(n) for n = 1..1000

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128402, A128403, A128404.

Select[Range[50000],Divisible[21^#-1,#^2]&] (* Harvey P. Dale, Jun 17 2011 *)

A128403 Numbers k such that k^2 divides 23^k-1.

Original entry on oeis.org

1, 2, 4, 6, 8, 11, 12, 20, 22, 24, 40, 42, 44, 60, 66, 78, 84, 88, 120, 132, 156, 168, 212, 220, 264, 312, 420, 424, 440, 444, 462, 474, 546, 620, 636, 660, 780, 820, 840, 858, 888, 924, 948, 1014, 1060, 1092, 1218, 1220, 1240, 1272, 1320, 1560, 1640, 1716

Offset: 1

Alexander Adamchuk, Mar 01 2007

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128404.

Join[{1},Select[Range[2000],PowerMod[23,#,#^2]==1&]] (* Harvey P. Dale, May 02 2015 *)

A128405 Numbers k such that k^2 divides 12^k - 1.

Original entry on oeis.org

1, 11, 253, 11891, 768361, 36112967, 61488361, 154261943, 2936791979, 61057324141, 67546215517, 107342336783, 186740152357, 347036920549, 429306186947, 468493520891, 635974117823, 797688507253, 3174672129299

Offset: 1

Alexander Adamchuk, Mar 01 2007

11 divides all except the first term.

All terms are congruent to +-1 (mod 12). - Robert G. Wilson v, Dec 19 2014

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

fQ[n_] := Mod[ PowerMod[12, n, n^2] - 1, n^2] == 0; k = 1; lst = {}; While[k < 100000001, If[ fQ@ k, AppendTo[lst, k]; Print@ k]; k += 10; If[ fQ@ k, AppendTo[lst, k]; Print@ k]; k += 2]; lst (* Robert G. Wilson v, Dec 19 2014 *)

a(6)-a(8) from Farideh Firoozbakht, Mar 05 2007

Terms a(9) onward from Max Alekseyev, May 06 2010

A129212 Numbers k such that k^3 divides 4^(k^2) - 1.

Original entry on oeis.org

1, 3, 21, 57, 219, 399, 903, 1533, 2667, 4161, 7077, 17157, 18543, 29127, 32547, 50673, 65919, 74109, 96141, 113799, 114681, 134463, 194691, 227829, 304311, 352317, 383907, 389193, 463071, 516621, 672987, 797349, 863517, 898779, 932799, 1252461, 1353639

Offset: 1

Alexander Adamchuk, Apr 03 2007

nonn

From Robert Israel, Aug 13 2020: (Start)
Except for 1, all terms are divisible by 3, but not 5 or 9.
All terms > 3 are divisible by at least one of 7, 19 and 73.
Are all terms squarefree? (End)

Robert Israel, Table of n, a(n) for n = 1..500 (first 49 terms from Robert Price)

Cf. A014945 (numbers k such that k divides 4^k-1).
Cf. A127104 (numbers k such that k^2 divides 4^k-1).
Cf. A128678 (numbers k such that k^3 divides 4^(k^2)+1).

filter:= n -> 4&^(n^2)-1 mod (n^3) = 0:
select(filter, [1,seq(i,i=3..10^6,6)]); # Robert Israel, Aug 13 2020

k=2; Do[ p=Prime[k]; If[ IntegerQ[ (PowerMod[ p+1, n^2, n^3 ] - 1 )/n^3 ], Print[ {k, p, n} ]], {n,1,200000} ]
k=2; Do[ p=Prime[k]; If[ IntegerQ[ (PowerMod[ p+1, n^2, n^3 ] - 1 )/n^3 ], Print[ {k, p, n} ]], {n,1000000} ] (* Robert G. Wilson v, Apr 06 2007 *)
Join[{1}, Select[Range[3000000], PowerMod[4, #^2, #^3] == 1 &]] (* Robert Price, Mar 31 2020 *)

More terms from Robert G. Wilson v, Apr 06 2007

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

Original entry on oeis.org

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

cp's OEIS Frontend

A128394 Numbers k such that k^2 divides 14^k - 1.

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A128396 Numbers k such that k^2 divides 16^k-1.

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A177917 Numbers k such that k^3 divides 17^(k^2) - 1.

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A128401 Numbers k such that k^2 divides 21^k-1.

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A128403 Numbers k such that k^2 divides 23^k-1.

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A128405 Numbers k such that k^2 divides 12^k - 1.

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A129212 Numbers k such that k^3 divides 4^(k^2) - 1.

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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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