A067946 -id:A067946 - cp's OEIS frontend

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

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

A015891 Numbers k such that k | 5^k + 5.

Original entry on oeis.org

1, 2, 5, 6, 10, 30, 70, 1565, 2806, 3126, 51670, 58290, 214405, 285286, 378258, 1854766, 2170486, 2222122, 2247610, 3463230, 4147522, 5942526, 9381126, 14818486, 15743890, 20162858, 34087054, 34838686, 38742166, 71067430

Offset: 1

Robert G. Wilson v

nonn

Giovanni Resta, Table of n, a(n) for n = 1..100

Cf. A067946 = numbers n such that n divides 5^n-1. Cf. A015951 = numbers n such that n | 5^n + 1.

Cf. A006517, A015888, A015889, A015892, A015893, A015897, A015898, A015902, A015903, A015904, A015905, A015906.

Select[Range[1000000], IntegerQ[(PowerMod[5,#,# ]+5)/# ]&] (* Alexander Adamchuk *)

Corrected by Alexander Adamchuk, Nov 04 2006

A123047 Numbers k that divide 5^k + 4.

Original entry on oeis.org

1, 3, 129, 60767, 76433163, 454034821, 26675718567, 164304369911289

Offset: 1

Alexander Adamchuk, Nov 04 2006

k must be odd since any power of 5 plus 4 is odd. - Robert G. Wilson v, Nov 14 2006
a(9) > 10^15. - Max Alekseyev, Oct 17 2016
Large term (may not be the next one): 3014733401203184049549. - Max Alekseyev, Oct 18 2013

Solutions to 5^n == k (mod n): A067946 (k=1), A015951 (k=-1), A124246 (k=2), A123062 (k=-2), A123061 (k=3), A123052 (k=-3), A125949 (k=4), this sequence (k=-4), A123091 (k=5), A015891 (k=-5), A277350 (k=6), A277348 (k=-6).

Do[ If[ PowerMod[5, 2n - 1, 2n - 1] + 5 == 2n, Print[2n - 1]], {n, 10^9}] (* Robert G. Wilson v *)

is(n)=Mod(5,n)^n==-4 \\ Charles R Greathouse IV, Nov 04 2016

a(4) and a(5) from Robert G. Wilson v, Nov 14 2006
a(7) from Ryan Propper, Mar 23 2007
a(8) from Max Alekseyev, Oct 17 2016

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 *)

A014960 Integers n such that n divides 24^n - 1.

Original entry on oeis.org

1, 23, 529, 1081, 12167, 24863, 50807, 279841, 571849, 1168561, 2387929, 2870377, 6436343, 7009273, 13152527, 15954479, 26876903, 54922367, 66018671, 112232663, 134907719, 148035889, 161213279, 302508121, 329435831

Offset: 1

Olivier Gérard

nonn

Also, numbers n such that n divides s(n), where s(1)=1, s(k)=s(k-1)+k*24^(k-1) (cf. A014942).
All n > 1 in the sequence are multiple of 23. - Conjectured by Thomas Baruchel, Oct 10 2003; proved by Max Alekseyev, Nov 16 2019
If n is a term and prime p|(24^n - 1), then n*p is a term. In particular, if n is a term and prime p|n, then n*p is a term. The smallest term with 3 distinct prime factors is a(16) = 15954479 = 23 * 47 * 14759. - Max Alekseyev, Nov 16 2019

Max Alekseyev, Table of n, a(n) for n = 1..146

Prime factors are listed in A087807.
Cf. A014942.
Integers n such that n divides b^n - 1: A067945 (b=3), A014945 (b=4), A067946 (b=5), A014946 (b=6), A067947 (b=7), A014949 (b=8), A068382 (b=9), A014950 (b=10), A068383 (b=11), A014951 (b=12), A116621 (b=13), A014956 (b=14), A177805 (b=15), A014957 (b=16), A177807 (b=17), A128358 (b=18), A125000 (b=19), A128360 (b=20), A014959 (b=22).

s = 1; Do[ If[ Mod[ s, n ] == 0, Print[n]]; s = s + (n + 1)*24^n, {n, 1, 100000}]
Join[{1},Select[Range[330*10^6],PowerMod[24,#,#]==1&]] (* Harvey P. Dale, Jan 19 2023 *)

More terms from Robert G. Wilson v, Sep 13 2000
a(9)-a(12) from Thomas Baruchel, Oct 10 2003
Edited and terms a(13) onward added by Max Alekseyev, Nov 16 2019

A123062 Numbers k that divide 5^k + 2.

Original entry on oeis.org

1, 7, 51373, 78127, 138943, 620299, 2842933, 137422693, 2259290321, 413879131637, 434757575329, 915535274009, 14864856896743

Offset: 1

Alexander Adamchuk, Nov 04 2006

No further terms up to 10^15. Larger term: 64629734103979763971. - Max Alekseyev, Oct 15 2016

Solutions to 5^n == k (mod n): A067946 (k=1), A015951 (k=-1), A124246 (k=2), this sequence (k=-2), A123061 (k=3), A123052 (k=-3), A125949 (k=4), A123047 (k=-4), A123091 (k=5), A015891 (k=-5), A277350 (k=6), A277348 (k=-6).

Select[Range[1000000], IntegerQ[(PowerMod[5,#,# ]+2)/# ]&]

is(n)=Mod(5,n)^n==-2 \\ Charles R Greathouse IV, Nov 04 2016

More terms from Farideh Firoozbakht, Nov 18 2006
a(9) from Ryan Propper, Jan 29 2007
a(10)-a(13) from Max Alekseyev, Jul 28 2009, Oct 15 2016

A123091 Numbers k such that k divides 5^k - 5.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 10, 11, 13, 15, 17, 19, 20, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 124, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 190, 191, 193, 197, 199, 211, 217, 223, 227, 229, 233, 239

Offset: 1

Alexander Adamchuk, Nov 04 2006

nonn

All primes are the terms of a(n). Nonprimes in a(n) are listed in A122782(n) = {1,4,10,15,20,65,124,190,217,310,435,561,781,...}. All pseudoprimes to base 5 are the terms of a(n). They are listed in A005936(n) = {4,124,217,561,781,...}. Numbers n up to 10^6 such that n divides 5^n + 5 are {1,2,5,6,10,30,70,1565,2806,3126,51670,58290,214405,285286,378258}.

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

Cf. A122782 (nonprimes n such that 5^n==5 (mod n)).
Cf. A005936 (pseudoprimes to base 5).
Cf. A067946 (numbers n such that n divides 5^n-1).
Cf. A015951 (numbers n such that n | 5^n + 1).

Select[Range[1000], IntegerQ[(PowerMod[5,#,# ]-5)/# ]&]

is(n)=Mod(5,n)^n==5 \\ Charles R Greathouse IV, Nov 04 2016

A124246 Numbers k that divide 5^k - 2.

Original entry on oeis.org

1, 3, 123, 202884639, 242405133, 92273577267, 2670733723929, 81035221987959

Offset: 1

Farideh Firoozbakht, Nov 19 2006

nonn

No other terms below 10^15. Some larger terms: 60092749466423900486673922957841, 401021769827858799355246286337987697472836927856337282726789534497163. - Max Alekseyev, Oct 15 2016

Solutions to 5^n == k (mod n): A067946 (k=1), A015951 (k=-1), this sequence (k=2), A123062 (k=-2), A123061 (k=3), A123052 (k=-3), A125949 (k=4), A123047 (k=-4), A123091 (k=5), A015891 (k=-5), A277350 (k=6), A277348 (k=-6).

Do[If[Mod[(PowerMod[5,n,n]-2),n]==0,Print[n]],{n,1000000000}]

is(n)=Mod(5,n)^n==2 \\ Charles R Greathouse IV, Nov 04 2016

a(6)-a(8) from Max Alekseyev, Jul 28 2009, Jun 02 2010, Oct 15 2016

A014956 Positive integers k such that k divides 14^k - 1.

Original entry on oeis.org

1, 13, 169, 2041, 2197, 26533, 28561, 114413, 320437, 344929, 371293, 1487369, 4165681, 4484077, 4826809, 17962841, 19335797, 24355253, 50308609, 54153853, 58293001, 62748517, 77457601, 233516933, 249302027, 251365361, 316618289

Offset: 1

Olivier Gérard

nonn

Also, positive integers k such that k divides A014929(k).
13 divides a(n) for n > 1. All powers of 13 are terms. All a(n) that are not powers of 13 are divisible either by 157 or 677 or both. - Alexander Adamchuk, May 14 2010
Prime divisors of a(n) in order of appearance: {13, 157, 677, 11933, 122147, 52807, ...}. - Alexander Adamchuk, May 16 2010

Cf. A067945, A014945, A067946, A014946, A067947, A014949, A068382, A014950, A068383, A014951, A116621, A177805, A014957, A177807, A128358, A128360.

Cf. A177914, A128394.

Join[{1}, Select[Range[2000000], PowerMod[14, #, #] == 1 &]] (* Robert Price, Mar 31 2020 *)

2 more terms from R. J. Mathar, Mar 05 2008
a(8)-a(23) from Alexander Adamchuk, May 14 2010
a(24)-a(44) from Alexander Adamchuk, May 16 2010
Edited by Max Alekseyev, Sep 10 2011

A123061 Numbers k that divide 5^k - 3.

Original entry on oeis.org

1, 2, 22, 77, 242, 371, 16102, 45727, 73447, 81286, 112277, 368237, 10191797, 13563742, 30958697, 389974222, 6171655457, 55606837682, 401469524477, 434715808966, 1729670231597, 12399384518278, 28370781933478, 32458602019394, 45360785149757, 1073804398767214

Offset: 1

Alexander Adamchuk, Nov 04 2006

nonn

Some larger terms: 10157607413638637338691, 678641208236297002873422185407157785099272404809011007522511134591325167. - Max Alekseyev, Oct 20 2016

Cf. A050259, A130422, A277554, A116629.

Solutions to 5^n == k (mod n): A067946 (k=1), A015951 (k=-1), A124246 (k=2), A123062 (k=-2), this sequence (k=3), A123052 (k=-3), A125949 (k=4), A123047 (k=-4), A123091 (k=5), A015891 (k=-5), A277350 (k=6), A277348 (k=-6).

Select[Range[1000000], IntegerQ[(PowerMod[5,#,# ]-3)/# ]&]
Do[If[IntegerQ[(PowerMod[5, n, n ]-3)/n], Print[n]], {n, 10^9}] (* Ryan Propper, Dec 30 2006 *)

is(n)=Mod(5,n)^n==3 \\ Charles R Greathouse IV, Nov 04 2016

More terms from Farideh Firoozbakht, Nov 18 2006
Corrected and extended by Ryan Propper, Jan 01 2007
Entry revised by N. J. A. Sloane, Jan 24 2007
a(18) from Lars Blomberg, Dec 12 2011
a(19)-a(26) from Max Alekseyev, Oct 20 2016

cp's OEIS Frontend

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

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A015891 Numbers k such that k | 5^k + 5.

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A123047 Numbers k that divide 5^k + 4.

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

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A014960 Integers n such that n divides 24^n - 1.

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A123062 Numbers k that divide 5^k + 2.

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A123091 Numbers k such that k divides 5^k - 5.

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A124246 Numbers k that divide 5^k - 2.

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