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A385224 Primes p such that multiplicative order of -4 modulo p is odd.

%I A385224 #20 Jun 28 2025 12:53:59
%S A385224 5,13,29,37,41,53,61,101,109,113,137,149,157,173,181,197,229,269,277,
%T A385224 293,313,317,349,373,389,397,409,421,457,461,509,521,541,557,569,593,
%U A385224 613,653,661,677,701,709,733,757,761,773,797,809,821,829,853,857,877,941,953,997
%N A385224 Primes p such that multiplicative order of -4 modulo p is odd.
%C A385224 The multiplicative order of -4 modulo a(n) is A385230(n).
%C A385224 Different from A133204: 593 is here but not in A133204, and 1601 is in A133204 but not here.
%C A385224 The sequence contains no primes congruent to 3 modulo 4 and all primes congruent to 5 modulo 8:
%C A385224   - If p is a term of this sequence, then -4 is a quadratic residue modulo p, so p == 1 (mod 4);
%C A385224   - For p == 1 (mod 4), we have (-4)^((p-1)/4) == (+-1+-i)^(p-1) == 1 (mod p), where i is a solution to i^2 == -1 (mod p).
%C A385224 Conjecture: this sequence has density 1/3 among the primes.
%H A385224 Jianing Song, <a href="/A385224/b385224.txt">Table of n, a(n) for n = 1..10000</a>
%t A385224 Select[Prime[Range[200]], OddQ[MultiplicativeOrder[-4, #]] &] (* _Paolo Xausa_, Jun 28 2025 *)
%o A385224 (PARI) isA385224(p) = isprime(p) && (p!=2) && znorder(Mod(-4,p))%2
%Y A385224 Subsequence of A002144 (primes congruent to 1 modulo 4).
%Y A385224 Contains A007521 (primes congruent to 5 or modulo 8) as a proper subsequence.
%Y A385224 Cf. A385230 (the actual multiplicative orders).
%Y A385224 Cf. other bases: A014663 (base 2), A385220 (base 3), A385221 (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), this sequence (base -4), A385225 (base -5).
%Y A385224 Cf. A133204.
%K A385224 nonn,easy
%O A385224 1,1
%A A385224 _Jianing Song_, Jun 22 2025