A385223 Primes p such that multiplicative order of -3 modulo p is odd.

2, 7, 19, 31, 37, 43, 61, 67, 79, 103, 127, 139, 151, 157, 163, 199, 211, 223, 271, 283, 307, 331, 349, 367, 373, 379, 397, 439, 463, 487, 499, 523, 547, 571, 607, 613, 619, 631, 643, 661, 691, 727, 739, 751, 787, 811, 823, 853, 859, 877, 883, 907, 919, 937, 967, 991, 997

Offset: 1

Author

Jianing Song, Jun 22 2025

Keywords

• nonn
• easy

Comments

The multiplicative order of -3 modulo a(n) is A385229(n).

Without 2, contained in primes congruent to 1 modulo 3 (primes p such that -3 is a quadratic residue modulo p, A002476), and contains primes congruent to 7 modulo 12 (A068229).

Conjecture: this sequence has density 1/3 among the primes.

Links

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

Crossrefs

Subsequence of A002476. Contains A068229 as a subsequence.

Cf. A385229 (the actual multiplicative orders).

Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), this sequence (base -3), A385224 (base -4), A385225 (base -5).

Programs

Mathematica

Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-3, #]] &] (* Paolo Xausa, Jun 28 2025 *)

PARI

isA385223(p) = isprime(p) && (p!=3) && znorder(Mod(-3,p))%2