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

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

1,1

Comments

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.

Links

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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).

Sequence in context: A386022 A385776 A163997 * A067931 A186267 A067660

Adjacent sequences: A385189 A385190 A385191 * A385193 A385194 A385195

Keyword

nonn,easy

Author

Jianing Song, Jun 20 2025

Status

approved