A127103 -id:A127103 - cp's OEIS frontend

A127106 Numbers k such that k^2 divides 6^k-1.

1, 5, 1555, 9673655, 187159211791705, 776119592182705

Offset: 1

Alexander Adamchuk, Jan 05 2007

Subsequence of A014946.

Cf. A127100, A127101, A127102, A127103, A127104, A127105, A127107, A127092, A014946.

Select[Range[20000], IntegerQ[(PowerMod[6, #, #^2 ]-1)/#^2 ]&]

Two more terms from Max Alekseyev, May 05 2010

A127102 Numbers k such that k^2 divides 8^k-1.

Original entry on oeis.org

1, 7, 889, 2359, 299593, 2033143, 13549249, 42931441, 100170217, 310935751, 685169191, 3606045247, 4566096913, 5452293007, 6620620783, 12721617559, 30987132463, 65576560063, 92349997537, 104785348087, 457967746369

Offset: 1

Alexander Adamchuk, Jan 05 2007, Jan 07 2007

nonn

Subsequence of A014949.
a(5) = 299593 = (8^7-1)/7 is a repunit in base 8, 1111111(base 8). The first 5 listed terms of a(n) are terms of the finite sequence A003530: divisors of 8^7-1. [Corrected and edited by M. F. Hasler, Nov 21 2018]
Also a subsequence of A177908, see there for more comments on properties of terms of this sequence. - M. F. Hasler, Nov 21 2018

Cf. A127100, A127101, A127103, A127104, A127105, A127106, A127107, A127092, A014949, A003530.

Select[Range[20000], IntegerQ[(PowerMod[8, #, #^2 ]-1)/#^2 ]&]

is(n)=Mod(8,n^2)^n==1 \\ M. F. Hasler, Nov 21 2018

More terms from Ryan Propper, Jan 05 2007

Terms a(12) onward from Max Alekseyev, May 06 2010

A127100 Numbers k such that k^2 divides 10^k-1.

Original entry on oeis.org

1, 3, 9, 111, 333, 3003003, 111111111, 225121209, 675363627, 27486820443, 32119664517, 82460461329, 24048075051027, 90180273183093, 225346555330209, 889778776887999, 3336670107774441, 10717272100393839, 19885751580714849, 27514334750263443

Offset: 1

Alexander Adamchuk, Jan 05 2007, Jan 07 2006

nonn

Subsequence of A014950.
First 7 terms are palindromes. a(n) is divisible by 3 for 1Alexander Adamchuk, Jan 25 2007
Except for 3, also numbers n such that the decimal expansion of 1/n^2 has period n. - Arkadiusz Wesolowski, Mar 13 2012

Cf. A127101, A127102, A127103, A127104, A127105, A127106, A127107, A127092, A014950.

Select[Range[20000], IntegerQ[(PowerMod[10, #, #^2 ]-1)/#^2 ]&]

More terms from Ryan Propper, Jan 06 2007

Further terms and edited by Max Alekseyev, May 09 2010

A127101 Numbers k such that k^2 divides 9^k - 1.

Original entry on oeis.org

1, 2, 4, 8, 10, 20, 40, 110, 136, 164, 220, 328, 440, 610, 680, 820, 1210, 1220, 1544, 1640, 2420, 2440, 2530, 4840, 5060, 5576, 6710, 7370, 7480, 7720, 9020, 10120, 11810, 13420, 13612, 14008, 14740, 18040, 18632, 19580

Offset: 1

Alexander Adamchuk, Jan 05 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3000 (terms 1..56 from R. J. Mathar)

Subset of A068382 (numbers k such that k divides 9^k - 1).

Cf. A127100, A127102, A127103, A123104, A127105, A127106, A127107, A127092.

Select[Range[20000], IntegerQ[(PowerMod[9, #, #^2 ]-1)/#^2 ]&]

is(k) = Mod(9, k^2)^k == 1; \\ Amiram Eldar, May 21 2024

A127105 Numbers k such that k^2 divides 5^k-1.

Original entry on oeis.org

1, 2, 4, 6, 12, 42, 52, 84, 156, 186, 372, 1092, 1218, 1302, 1806, 2436, 2604, 2756, 3612, 4836, 5334, 7212, 8268, 10668, 12324, 15918, 18858, 24492, 31668, 31836, 33852, 37716, 37758, 46956, 50484, 52374, 55986, 57876, 71862, 75516, 86268

Offset: 1

Alexander Adamchuk, Jan 05 2007

nonn

Subset of A067946 (numbers k such that k divides 5^k-1).

Robert Israel, Table of n, a(n) for n = 1..1000
Taro Morishima, Über die Fermatsche Vermutung, Proc. Imp. Acad., Volume 4, Number 10 (1928), 590-592.

Cf. A127100, A127101, A127102, A127103, A123104, A127106, A127107, A127092.

Cf. A067946 (numbers k such that k divides 5^k-1).

select(t -> (5 &^t - 1) mod (t^2) = 0, [$1..10^5]); # Robert Israel, Jul 15 2018

Select[Range[30000], IntegerQ[(PowerMod[5, #, #^2 ]-1)/#^2 ]&]

isok(n) = Mod(5, n^2)^n == 1; \\ Michel Marcus, Apr 23 2017

More terms from Ryan Propper and Alexander Adamchuk, Jan 05 2007

A127107 Numbers n such that n^2 divides 7^n-1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 20, 24, 40, 57, 60, 100, 114, 120, 156, 200, 220, 228, 258, 300, 312, 440, 456, 516, 600, 660, 780, 1032, 1100, 1140, 1320, 1560, 1640, 1752, 1860, 2172, 2200, 2280, 2580, 2964, 3300, 3660, 3720, 3820, 3900, 4344, 4632, 4902, 4920, 5060

Offset: 1

Alexander Adamchuk, Jan 05 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A127100, A127101, A127102, A127103, A123104, A127105, A127106, A127092. Cf. A067947 = numbers n such that n divides 7^n-1.

Select[Range[10000], IntegerQ[(PowerMod[7, #, #^2 ]-1)/#^2 ]&]

A127104 Numbers k such that k^2 divides 4^k-1.

Original entry on oeis.org

1, 3, 21, 903, 2667, 7077, 113799, 114681, 304311, 389193, 898779, 932799, 4893357, 6099429, 8131683, 8776257, 14452473, 38350263, 38647497, 40647747, 49427511, 99583113, 118465473, 128794323, 131158041, 152643813, 262275447, 300510651, 314353263, 335873559, 349662369

Offset: 1

Alexander Adamchuk, Jan 05 2007

nonn

From Alexander Adamchuk, Jan 11 2007: (Start)
3 divides a(n) for n > 1.
7 divides a(n) for n > 2.
43 divides a(n) for n = {4, 8, 9, 10, 12, 13, 16, ...}.
127 divides a(n) for n = {5, 8, 11, 14, 15, 17, ...}.
Prime factors of a(n) in order of their appearance in {a(n)} are {3, 7, 43, 127, 337, 5419, 431, 1033, 5419, 2287, 3049, 9719, ...}. (End)

Amiram Eldar, Table of n, a(n) for n = 1..56

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

Subset of A014945 (numbers k such that k divides 4^nk-1).

Select[Range[30000], IntegerQ[(PowerMod[4, #, #^2 ]-1)/#^2 ]&]

is(k) = Mod(4, k^2)^k == 1; \\ Amiram Eldar, May 25 2024

More terms from Ryan Propper and Alexander Adamchuk, Jan 05 2007

a(27)-a(31) from Amiram Eldar, May 25 2024

A128393 Numbers k such that k^2 divides 13^k - 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 12, 14, 20, 28, 42, 60, 68, 84, 140, 183, 204, 220, 340, 366, 406, 420, 476, 660, 732, 812, 942, 1020, 1218, 1428, 1540, 1806, 1860, 1884, 2380, 2436, 2562, 3612, 3660, 3740, 4060, 4620, 5060, 5124, 6594, 7004, 7140, 8420, 9420, 9940, 11220

Offset: 1

Alexander Adamchuk, Mar 01 2007

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

Join[{1},Select[Range[12000],PowerMod[13,#,#^2]==1&]] (* Harvey P. Dale, Sep 05 2012 *)

A128397 Numbers k such that k^2 divides 17^k-1.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 36, 40, 42, 48, 60, 72, 78, 80, 84, 116, 120, 126, 144, 156, 168, 180, 220, 232, 234, 240, 252, 312, 336, 342, 348, 360, 420, 440, 464, 468, 504, 546, 580, 624, 660, 684, 696, 720, 780, 840, 880, 936, 1008, 1044, 1092, 1160

Offset: 1

Alexander Adamchuk, Mar 01 2007

The first 9 numbers (and many more) which are a multiple of their digital sum in all bases from 2 through 16 (A218087), are members of this sequence, and also of A177917. - M. F. Hasler, Oct 22 2012

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

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

Cf. A177917.

is_A128397(n)=Mod(17,n^2)^n==1  \\ M. F. Hasler, Oct 22 2012

A128398 Numbers k such that k^2 divides 18^k - 1.

Original entry on oeis.org

1, 17, 128583032925805678351

Offset: 1

Alexander Adamchuk, Mar 08 2007

17 divides a(n) for n > 1.

a(4) > 10^50. - Max Alekseyev, Mar 01 2025

Subsequence of A128358.

Cf. A127103, A127104, A127105, A127106, A127107, A127102, A127101, A127100, A127092, A128405, A128393, A128394, A128395, A128396, A128397, A128398, A128399, A128400, A128401, A128402, A128403, A128404.

cp's OEIS Frontend

A127106 Numbers k such that k^2 divides 6^k-1.

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A127102 Numbers k such that k^2 divides 8^k-1.

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A127100 Numbers k such that k^2 divides 10^k-1.

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A127101 Numbers k such that k^2 divides 9^k - 1.

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A127105 Numbers k such that k^2 divides 5^k-1.

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A127107 Numbers n such that n^2 divides 7^n-1.

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A127104 Numbers k such that k^2 divides 4^k-1.

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A128393 Numbers k such that k^2 divides 13^k - 1.

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Mathematica

A128397 Numbers k such that k^2 divides 17^k-1.

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