A127102 -id:A127102 - cp's OEIS frontend

A177916 Numbers k such that k^3 divides 16^(k^2) - 1.

1, 3, 5, 15, 21, 39, 55, 57, 105, 111, 155, 165, 195, 205, 219, 273, 285, 327, 399, 465, 505, 555, 609, 615, 741, 777, 903, 915, 1095, 1155, 1255, 1265, 1365, 1443, 1515, 1533, 1635, 1705, 1995, 2067, 2109, 2145, 2255, 2265, 2289, 2373, 2667, 2715, 2847

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

From Robert Israel, Apr 24 2015: (Start)
The only primes in the sequence are 3 and 5.
If m and n are in the sequence and are coprime, then m*n is in the sequence.
Are all members of the sequence divisible by 3 or 5? (End)

Robert Israel, Table of n, a(n) for n = 1..1000

Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.

Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

[n: n in [1..2000] | Denominator((16^(n^2) - 1)/n^3) eq 1]; // Vincenzo Librandi, Apr 24 2015

A177916:=n->`if`((16&^(n^2)-1) mod n^3 = 0, n, NULL): seq(A177916(n), n=1..1000); # Wesley Ivan Hurt, Apr 23 2015

Select[Range[3000], IntegerQ[(16^(#^2) - 1) / #^3 ] &] (* Vincenzo Librandi, Apr 24 2015 *)
Join[{1},Select[Range[3000],PowerMod[16,#^2,#^3]==1&]] (* Harvey P. Dale, Dec 05 2024 *)

A177920 Numbers k such that k^3 divides 20^(k^2) - 1.

Original entry on oeis.org

1, 19, 7805561, 1432001198261, 3661225986659, 58130944174609, 187470481770989

Offset: 1

Alexander Adamchuk, May 14 2010

19 divides a(n) for n > 1.

Cf. A128360 (k divides 20^k-1), A128400 (k^2 divides 20^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[10000000], PowerMod[20, #^2, #^3] == 1 &]] (* Robert Price, Apr 04 2020 *)

a(4)-a(7) from Max Alekseyev, Oct 02 2010

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

A177905 Numbers k such that k^3 divides 5^(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 6, 12, 26, 42, 52, 68, 78, 84, 114, 156, 186, 204, 222, 228, 372, 444, 546, 798, 876, 884, 1092, 1218, 1252, 1302, 1378, 1428, 1482, 1554, 1596, 1806, 2418, 2436, 2604, 2652, 2756, 2886, 2964, 3108, 3534, 3606, 3612, 3756, 3876, 4134, 4218, 4836

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..900

Cf. A067946 (k divides 5^k-1), A127105 (k^2 divides 5^k-1).
Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.
Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

A177905:=n->`if`(5^(n^2)-1 mod n^3 = 0, n, NULL): seq(A177905(n), n=1..10^3); # Wesley Ivan Hurt, Feb 16 2017

Join[{1},Select[Range[5000],PowerMod[5,#^2,#^3]==1&]] (* Harvey P. Dale, May 04 2017 *)

A177907 Numbers k such that k^3 divides 7^(k^2) - 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 20, 24, 30, 40, 50, 57, 60, 68, 78, 100, 110, 111, 114, 120, 136, 150, 156, 200, 204, 220, 222, 228, 258, 300, 312, 330, 340, 390, 408, 440, 444, 456, 516, 550, 570, 600, 660, 680, 780, 820, 876, 888, 930, 1010, 1020, 1032, 1086, 1100

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

Robert Price, Table of n, a(n) for n = 1..2835

Cf. A067947 (k divides 7^k-1), A127107 (k^2 divides 7^k-1).
Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.
Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

Select[Range[1100],Divisible[7^(#^2)-1,#^3]&] (* Harvey P. Dale, Nov 05 2011 *)

A177909 Numbers k such that k^3 divides 9^(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 8, 10, 20, 40, 68, 82, 110, 136, 164, 220, 328, 340, 410, 440, 610, 680, 772, 820, 1010, 1210, 1220, 1510, 1544, 1640, 2020, 2420, 2440, 2530, 2788, 3020, 3740, 3860, 4040, 4510, 4840, 5060, 5576, 6040, 6710, 6806, 7004, 7370, 7480, 7720, 8020, 9020

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

			9^(2^2) - 1 = 6560, which is divisible by 2^3, so 2 is in the sequence.
9^(4^2) - 1 = 1853020188851840, which is divisible by 4^3, so 4 is in the sequence.
9^(6^2) - 1 = 22528399544939174411840147874772640, which is not divisible by 6, and certainly not by 6^3, so 6 is not in the sequence.

Robert Price, Table of n, a(n) for n = 1..446 (terms 1..119 from R. J. Mathar).

Cf. A068382 (k divides 9^k-1), A127101 (k^2 divides 9^k-1).
Cf. A129211, A129212, A177905, A127106, A177907, A127102, A177909, A177243.
Cf. A177911, A177912, A177913, A177914, A177915, A177916, A177917, A177918, A177919, A177920.

A177909:=n->`if`(9^(n^2)-1 mod n^3 = 0,n,NULL): seq(A177909(n), n=1..1000); # Wesley Ivan Hurt, Oct 04 2014

Select[Range[100], Divisible[9^(#^2) - 1, #^3] &] (* Alonso del Arte, Oct 04 2014 *)

A177911 Numbers k such that k^3 divides 11^(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 20, 30, 42, 60, 68, 78, 84, 114, 122, 156, 204, 210, 222, 228, 244, 340, 366, 390, 420, 444, 546, 570, 610, 732, 780, 798, 820, 876, 930, 1010, 1020, 1092, 1110, 1140, 1164, 1218, 1220, 1428, 1482, 1554, 1596, 1806, 1830, 1860, 2020, 2220

Offset: 1

Alexander Adamchuk, May 14 2010

nonn

Robert Price, Table of n, a(n) for n = 1..1634

Cf. A068383 (k divides 11^k-1), A127092 (k^2 divides 11^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[2300],PowerMod[11,#^2,#^3]==1&]] (* Harvey P. Dale, Aug 24 2024 *)

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

cp's OEIS Frontend

A177916 Numbers k such that k^3 divides 16^(k^2) - 1.

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A177920 Numbers k such that k^3 divides 20^(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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A128404 Numbers k such that k^2 divides 24^k-1.

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A177905 Numbers k such that k^3 divides 5^(k^2) - 1.

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A177907 Numbers k such that k^3 divides 7^(k^2) - 1.

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A177909 Numbers k such that k^3 divides 9^(k^2) - 1.

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A177911 Numbers k such that k^3 divides 11^(k^2) - 1.

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