A385221 Primes p such that multiplicative order of 4 modulo p is odd.

3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 73, 79, 83, 89, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 233, 239, 251, 263, 271, 281, 283, 307, 311, 331, 337, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 563

Offset: 1

Author

Jianing Song, Jun 22 2025

Keywords

• nonn
• easy

Comments

The multiplicative order of 4 modulo a(n) is A385227(n).

Primes p such that neither ord(2,p) nor ord(-2,p) is divisible by 4, where ord(a,m) is the multiplicative order of a modulo m. (Note that we have either (a) ord(2,p) = ord(-2,p) and both are even; (b) ord(-2,p) = 2*ord(2,p), ord(2,p) is odd, ord(-2,p) == 2 (mod 4); or (c) ord(2,p) = 2*ord(-2,p), ord(-2,p) is odd, ord(2,p) == 2 (mod 4)).

Contains all primes congruent to 3 modulo 4 (A002145).

Conjecture: this sequence has density 7/12 among the primes (see A014663).

Links

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

Crossrefs

Contains A002145, A014663, and A163183.

Cf. A385227 (the actual multiplicative orders).

Cf. other bases: A014663 (base 2), A385220 (base 3), this sequence (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5).

Programs

Mathematica

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[4, #]] &] (* Paolo Xausa, Jun 28 2025 *)

PARI

isA385221(p) = isprime(p) && (p!=2) && znorder(Mod(4,p))%2