A093546 -id:A093546 - cp's OEIS frontend

A127263 Numbers k such that k^3 divides 2^(k^2)+1.

1, 3, 57, 32547, 9961491, 297381939, 1338104811, 3942759027, 5688011361, 8920514307, 9146532873, 40253706489, 243640690617, 764039295291, 1127102902923, 1556475424971, 2251315404417, 3005607686883, 5222670270483

Offset: 1

Max Alekseyev, Mar 27 2007, Mar 29 2007, Apr 18 2007

nonn

If k belongs to this sequence, then so does (2^(k^2)+1)/k^2.
From Alexander Adamchuk, May 14 2010: (Start)
3 divides a(n) for n>1.
19 divides a(n) for n>2. (End)

ArtOfProblemSolving forum, n^3 | (2^{n^2}+1)

Cf. A006521, A093546, A093665, A136372, A128677.

Cf. A128678, A128679, A128680, A128681, A128682, A128683, A128684, A128685, A177813, A177814, A177816, A177817, A177818, A177819, A177820.

Select[Range[100000], Divisible[2^(#^2) + 1, #^3] &] (* Robert Price, Mar 23 2020 *)

a(7) from Ryan Propper, Jan 01 2008

a(8)-a(19) from Max Alekseyev, May 14 2010

A093547 Numbers k such that k divides 3^(k^2) - 1.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 20, 32, 40, 50, 64, 68, 80, 100, 110, 128, 136, 160, 164, 200, 220, 250, 256, 272, 320, 328, 340, 400, 440, 500, 512, 544, 550, 610, 640, 656, 680, 772, 800, 820, 880, 1000, 1010, 1024, 1088, 1100, 1156, 1210, 1220, 1250, 1280, 1312, 1360

Offset: 1

Farideh Firoozbakht, Mar 31 2004

nonn

This sequence is closed under multiplication, i.e., if x and y are terms then so is x*y.

A067945 is a subsequence of this sequence. A067945 is also closed under multiplication. In fact if m is an integer and k is a natural number then the sequence " n divides m^(n^k) - 1 " is a subsequence of the sequence " n divides m^n^(k+1)- 1 " and both are closed under multiplication.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A067945, A093546, A006521.

v={};Do[If[IntegerQ[(3^n^2-1)/n], v=Append[v, n];Print[v]], {n, 2500}]
Select[Range[1400],Divisible[3^#^2-1,#]&] (* Harvey P. Dale, Nov 04 2015 *)

isok(k) = Mod(3, k)^(k^2) == 1; \\ Amiram Eldar, May 26 2024

A092408 Numbers k that divide 3^(k^3) + 1.

Original entry on oeis.org

1, 2, 34, 386, 578, 6562, 9826, 74498, 111554, 167042, 1100546, 1192354, 1266466, 1896418, 2839714, 14378114, 18709282, 20270018, 21529922, 32239106, 35759426, 48275138, 191812802, 212405378, 230124322, 244427938, 318057794, 344590306

Offset: 1

Robert G. Wilson v, Apr 02 2004

nonn

It appears that all numbers of the form 2*17^m are present.

Cf. A015949, A093546, A093666.

Select[ Range[ 350000000], PowerMod[3, #^3, # ] +1 == # &]

A093665 Numbers k that divide 2^(k^3) + 1.

Original entry on oeis.org

1, 3, 9, 27, 57, 81, 171, 243, 513, 729, 1083, 1467, 1539, 2187, 3249, 4401, 4617, 6561, 9747, 13131, 13203, 13851, 19683, 20577, 27873, 29241, 32547, 39393, 39609, 41553, 59049, 61731, 83619, 87723, 97641, 118179, 118827, 124659, 177147, 185193

Offset: 1

Robert G. Wilson v, Apr 03 2004

nonn

Cf. A006521, A093546.

Select[ Range[ 249488], PowerMod[2, #^3, # ] == # - 1 &]

A093666 Numbers n such that n | 3^n^2 + 1.

Original entry on oeis.org

1, 2, 82, 3362, 137842, 188354, 5651522, 7722514, 13232914, 231712402, 316623074, 432649138, 542549474, 1196468642, 2650762258, 9500208482, 12981546034, 17738614658, 22244528434, 30396003458, 49055214322, 108681252578, 389508547762, 532243387394, 727283200978, 912025665794, 993795069986

Offset: 1

Robert G. Wilson v, Apr 02 2004

nonn

If n is a term and p is its odd prime divisor, then p*n is also a term. In particular, the sequence contains 2*41^k and 2*41*2297^k for all k>=1.

Cf. A015949, A092408, A093546.

Select[ Range[ 250000000], PowerMod[3, #^2, # ] +1 == # & ]

Terms a(11) onward from Max Alekseyev, Feb 13 2012

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A127263 Numbers k such that k^3 divides 2^(k^2)+1.

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A093547 Numbers k such that k divides 3^(k^2) - 1.

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A092408 Numbers k that divide 3^(k^3) + 1.

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A093665 Numbers k that divide 2^(k^3) + 1.

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A093666 Numbers n such that n | 3^n^2 + 1.

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