A127100 -id:A127100 - 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

A127092 Numbers k such that k^2 divides 11^k - 1.

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 20, 30, 42, 60, 84, 114, 156, 210, 222, 228, 244, 420, 444, 570, 732, 780, 798, 930, 1092, 1110, 1140, 1220, 1554, 1596, 1806, 1860, 2220, 2436, 2964, 3108, 3612, 3660, 3990, 4218, 5124, 5460, 5772, 6510, 7770, 7980, 8268, 8436, 9030

Offset: 1

Alexander Adamchuk, Jan 05 2007

nonn

Subsequence of A068383.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=11 of A333500.

Cf. A068383, A127100, A127101, A127102, A127104, A127105, A127106, A127107, A292336.

Select[Range[15000], IntegerQ[(PowerMod[11, #, #^2 ]-1)/#^2 ]&]
Join[{1},Select[Range[9100],PowerMod[11,#,#^2]==1&]] (* Harvey P. Dale, Dec 30 2018 *)

for(k=1, 1e4, if(Mod(11, k^2)^k==1, print1(k", "))) \\ Seiichi Manyama, Mar 25 2020

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 ]&]

A014950 Numbers m such that m divides 10^m - 1.

Original entry on oeis.org

1, 3, 9, 27, 81, 111, 243, 333, 729, 999, 2187, 2997, 4107, 6561, 8991, 12321, 13203, 19683, 20439, 26973, 36963, 39609, 59049, 61317, 80919, 110889, 118827, 151959, 177147, 183951, 242757, 332667, 356481, 455877, 488511, 531441, 551853, 728271

Offset: 1

Olivier Gérard

nonn

Also, m such that m | R(m) = A002275(m). - Lekraj Beedassy, Mar 25 2005
For n > 1, 3 divides a(n). If m is in the sequence and d divides m then for each positive integer k, d^k*m is in the sequence. So if m is in the sequence then m^k is in the sequence for each positive integer k. In particular, 3^k is in this sequence for all k. - Farideh Firoozbakht, Apr 14 2010
Numbers m such that m divides s(m), where s(1) = 1, s(k) = s(k-1) + k*10^(k-1).
Number of terms <= 10^k, beginning with k = 0: 1, 3, 5, 10, 15, 25, 41, 68, 108, 178, 291, ... - Robert G. Wilson v, Nov 30 2013
Numbers m such that m divides A033713(m). - Hans Havermann, Jan 25 2014

J. D. E. Konhauser et al., Which Way Did The Bicycle Go? Problem 80 pp. 26; 133, Dolciani Math. Exp., No. 18, MAA, Washington DC, 1996.

Hans Havermann, Table of n, a(n) for n = 1..1600 (first 800 terms from Robert G. Wilson v)
C. Cooper and R. E. Kennedy, Niven Repunits and 10^n = 1 (mod n), The Fibonacci Quarterly, pp. 139-143, vol 27, May 02 1989.
Hans Havermann, A014950 factorized and atomized.

Cf. A066364, A114207, A122787, A127100, A232769.

Select[ Range[3, 1000000, 6], PowerMod[10, #, #] == 1 &] (* modified by Robert G. Wilson v, Dec 03 2013 *)
k = 3; A014950 = {1}; While[k < 1000000, If[ PowerMod[ 10, k, k] == 1, AppendTo[ A014950, k]; Print@ k]; k += 6]; A014950 (* Robert G. Wilson v, Nov 29 2013 *)

is(n)=Mod(10,n)^n==1 \\ Charles R Greathouse IV, Nov 29 2013

Solutions to 10^m == 1 (mod m). - Vladeta Jovovic

More terms from Vladeta Jovovic, Dec 18 2001
More terms from Larry Reeves (larryr(AT)acm.org), Jan 06 2005
Edited by Max Alekseyev, May 20 2011

A127103 Numbers k such that k^2 divides 3^k-1.

Original entry on oeis.org

1, 2, 4, 20, 220, 1220, 2420, 5060, 13420, 14740, 23620, 55660, 145420, 147620, 162140, 237820, 259820, 290620, 308660, 339020, 447740, 847220, 899140, 1210220, 1440820, 1599620, 1759340, 2332660, 2616020, 2858020, 3196820, 3344660

Offset: 1

Alexander Adamchuk, Jan 05 2007

nonn

From Alexander Adamchuk, Jan 11 2007: (Start)
2 divides a(n) for n>1. 2^2 divides a(n) for n>2. 5 divides a(n) for n>3.
11 divides a(n) for n = {5,7,8,9,10,12,13,14,15,16,17,18,19,20,22,23,24,26,27, 28,29,30,31,31,33,34,35,...}.
11^2 divides a(n) for n = {7,12,14,15,26,27,29,30,31,33,34,...}.
Prime factors of a(n) in order of their appearance in a(n) are {2,5,11,61,23,67,1181,661,47,1321,367,3851,5501,727,461,269,...}. (End)

Amiram Eldar, Table of n, a(n) for n = 1..300

Subset of A067945 (numbers k that divide 3^k - 1).

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

Select[Range[30000], IntegerQ[(PowerMod[3, #, #^2 ]-1)/#^2 ]&]
Join[{1},Select[Range[335*10^4],PowerMod[3,#,#^2]==1&]] (* Harvey P. Dale, Oct 02 2019 *)

is(k) = Mod(3, k^2)^k == 1; \\ Amiram Eldar, May 21 2024

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

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 *)

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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A127092 Numbers k such that k^2 divides 11^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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A014950 Numbers m such that m divides 10^m - 1.

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A127103 Numbers k such that k^2 divides 3^k-1.

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