A040105 -id:A040105 - cp's OEIS frontend

A122869 Primes p that divide Lucas((p-1)/2), where Lucas is A000032.

11, 19, 31, 59, 71, 79, 131, 139, 151, 179, 191, 199, 211, 239, 251, 271, 311, 331, 359, 379, 419, 431, 439, 479, 491, 499, 571, 599, 619, 631, 659, 691, 719, 739, 751, 811, 839, 859, 911, 919, 971, 991, 1019, 1031, 1039, 1051, 1091, 1151, 1171, 1231, 1259

Offset: 1

Alexander Adamchuk, Sep 16 2006

nonn

Final digit of a(n) is 1 or 9.
A002145 is the union of this sequence and A122870, Primes p that divide Lucas((p+1)/2).
Conjecture: This sequence is just the primes congruent to 11 or 19 mod 20. - Charles R Greathouse IV, May 25 2011 [The conjecture is correct. - Jianing Song, Jun 20 2025]
Note that F(p-1) = F((p-1)/2)*Lucas((p-1)/2), where F = A000045. Since gcd(F(n),Lucas(n)) = 1 or 2 (because Lucas(n)^2 - 5*F(n)^2 = 4*(-1)^n), this sequence lists primes p such that p divides F(p-1) but does not divides F((p-1)/2). By Propositions 1.1 and 1.2 (the k = 3 case) of my link below, this is primes p == 11, 19 (mod 20). - Jianing Song, Jun 20 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Jianing Song, Lucas sequences and entry point modulo p
Eric Weisstein's World of Mathematics, Gaussian Prime.
Eric Weisstein's World of Mathematics, Lucas Number.

Cf. A000032, A000045, A053032, A076518, A122870.

Subsequence of A002145, A003626, A040105, A040147 and A064739.

Select[Prime[Range[1000]],IntegerQ[(Fibonacci[(#1-1)/2-1]+Fibonacci[(#1-1)/2+1])/#1]&]

lista(kmax) = {my(lucas1 = 1, lucas2 = 3, lucas3, p); for(k = 3, kmax, lucas3 = lucas1 + lucas2; p = 2*k + 1; if(isprime(p) && !(lucas3 % p), print1(p, ", ")); lucas1 = lucas2; lucas2 = lucas3);} \\ Amiram Eldar, Jun 06 2024

A385192 Primes p such that multiplicative order of 5 modulo p is odd.

Original entry on oeis.org

2, 11, 19, 31, 59, 71, 79, 101, 109, 131, 139, 149, 151, 179, 181, 191, 199, 211, 239, 251, 269, 271, 311, 331, 359, 379, 389, 401, 409, 419, 431, 439, 461, 479, 491, 499, 541, 569, 571, 599, 619, 631, 659, 691, 719, 739, 751, 811, 829, 839, 859, 911, 919, 941, 971, 991

Offset: 1

Jianing Song, Jun 20 2025

The multiplicative order of 5 modulo a(n) is A385193(n).
Contained in primes congruent to 1 or 4 modulo 5 (primes p such that 5 is a quadratic residue modulo p, A045468), and contains primes congruent to 11 or 19 modulo 20 (A122869).
Conjecture: this sequence has density 1/3 among the primes.

			101 is a term since 5^25 == 1 (mod 101).

Jianing Song, Table of n, a(n) for n = 1..10000

Subsequence of A040105, which (without the terms 2 and 5) is itself a subsequence of A045468.
Contains A122869 as a proper subsequence.
Cf. A385193 (the actual multiplicative orders).
Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), this sequence (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[5, #]] &] (* Paolo Xausa, Jun 28 2025 *)

isA385192(p) = isprime(p) && (p!=5) && znorder(Mod(5,p))%2

A040106 Primes p such that x^4 = 5 has no solution mod p.

Original entry on oeis.org

3, 7, 13, 17, 23, 29, 37, 41, 43, 47, 53, 61, 67, 73, 83, 89, 97, 103, 107, 113, 127, 137, 157, 163, 167, 173, 193, 197, 223, 227, 229, 233, 241, 257, 263, 277, 281, 283, 293, 307, 313, 317, 337, 347, 349, 353, 367

Offset: 1

N. J. A. Sloane

Complement of A040105 relative to A000040. - Vincenzo Librandi, Sep 17 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

[p: p in PrimesUpTo(500) | not exists{x : x in ResidueClassRing(p) | x^4 eq 5} ]; // Vincenzo Librandi, Sep 17 2012

ok[p_]:= Reduce[Mod[x^4 - 5, p] == 0, x, Integers] == False;Select[Prime[Range[100]], ok] (* Vincenzo Librandi, Sep 17 2012  *)

cp's OEIS Frontend

A122869 Primes p that divide Lucas((p-1)/2), where Lucas is A000032.

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A385192 Primes p such that multiplicative order of 5 modulo p is odd.

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A040106 Primes p such that x^4 = 5 has no solution mod p.

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