1,1

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).

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

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

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

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).

Sequence in context: A160216 A181516 A285015 * A002145 A002052 A369249

Adjacent sequences: A385218 A385219 A385220 * A385222 A385223 A385224

nonn,easy

Jianing Song, Jun 22 2025

approved