A087807 -id:A087807 - cp's OEIS frontend

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

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

A164816 Prime factors in a divisibility sequence of the Lucas sequence v(P=3,Q=5) of the second kind.

Original entry on oeis.org

3, 17, 103, 163, 373, 487, 1733, 3469, 4373, 8803, 10259, 15607, 16069, 26237, 26297, 31193, 31517, 35153, 37987, 38047, 38149, 39367, 52817, 60427, 60589, 61553, 74357, 76837, 78713, 100733, 103979, 114377, 119891, 152189, 181277, 231131, 235891, 238307, 239783, 280927, 289243, 316903, 338581

Offset: 1

Jonathan Vos Post, Aug 26 2009

nonn

This is the last sequence on p. 15 of Smyth. [WARNING: Smyth lists 2 as a possible prime factor, which, in fact, is not possible. - Max Alekseyev, Sep 17 2024]
The Lucas sequence with P = 3, Q = 5 is defined as v=2,3,-1,-18,-49,-57,.. where v(n) = P*v(n-1)-Q*v(n-2), with g.f. (2-3x)/(1-3x+5x^2).
The indices n such that n|v(n) define the sequence T = 1,3,9,27,81,153,243,459,... as listed by Smyth.
The OEIS sequence shows all distinct prime factors of elements of T.

Max Alekseyev, Table of n, a(n) for n = 1..627
Richard André-Jeannin, Divisibility of generalized Fibonacci and Lucas numbers by their subscripts, Fibonacci Quart., 29(4) (1991) 364-366.
Yu. Bilu, G. Hanrot, and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539 (2001) 75-122.
R. D. Carmichael, On the numerical factors of the arithmetic forms alpha*n+-beta*n, Annals of Math., 2nd ser., 15 (1/4) (1913/14) 30-48.
Chris Smyth, The Terms in Lucas Sequences Divisible by their Indices, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.4. Preprint: arXiv:0908.3832 [math.NT], 2009.

Cf. A057719, A066364, A087807.

More detailed definition, comments rephrased, non-ascii characters in URL's removed - R. J. Mathar, Sep 09 2009
a(8)-a(9), a(11), a(18) from Jean-François Alcover, Dec 08 2017
Incorrect codes (depending on a search limit) removed, prime 2 removed, terms a(10), (12)-a(17), and a(19) onward added by Max Alekseyev, Sep 17 2024

A354026 Primes that divide some k dividing 4^k + 3^k (A045584).

Original entry on oeis.org

7, 379, 14407, 689431, 4235659, 41647747, 137534083, 239900179, 242121643, 349909477, 1245283747, 1478065891, 1605314383, 2500276549, 2748751303, 5618210347, 7490947129, 11236420693, 11260421089, 16948514941, 29440659361, 74163546829, 75093609319, 82188727303

Offset: 1

Max Alekseyev, May 15 2022

nonn

Prime p > 3 is in this sequence iff all prime factors of the multiplicative order of -3/4 modulo p belong to this sequence.

Cf. A023394, A045584, A066364, A087807, A129729, A134360, A171980, A354027.

S=[]; forprime(p=5,oo, f=Set(factor(znorder(Mod(-3/4,p)))[,1]); if(#setintersect(S,f)==#f, S=setunion(S,[p]); print1(p,", ")));

a(18)-a(24) from Jinyuan Wang, Jan 29 2025

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

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A164816 Prime factors in a divisibility sequence of the Lucas sequence v(P=3,Q=5) of the second kind.

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A354026 Primes that divide some k dividing 4^k + 3^k (A045584).

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