A385167 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) is even. Here ord(a,m) is the multiplicative order of a modulo m.

11, 79, 131, 199, 211, 239, 251, 331, 359, 439, 479, 491, 571, 599, 691, 719, 811, 839, 919, 971, 1039, 1051, 1091, 1171, 1279, 1291, 1319, 1399, 1439, 1451, 1531, 1559, 1571, 1759, 1811, 1879, 1931, 1999, 2011, 2039, 2131, 2239, 2251, 2371, 2399, 2411, 2531, 2719, 2731, 2851, 2879, 2971, 2999

Offset

1,1

Comments

Of course if a and m are integers, it doesn't matter if the base ring is Z or Z[i] for ord(a,m).

List of p = A002145(k) such that A385166(k) is even.

Since in this case d(p) divides (p^2-1)/2, 5 must be a quadratic residue modulo p (see A385165).

Links

Jianing Song, Table of n, a(n) for n = 1..10001

Example

359 is a term since the multiplicative order of 2+-i modulo 359 is 6444, and (360*ord(5,359))/6444 = 10 is even.

Prog

(PARI) quot(p) = my(z = znorder(Mod(5, p)), d = divisors((p+1)*z)); for(i=1, #d, if(Mod([2, -1; 1, 2], p)^d[i] == 1, return((p+1)*z/d[i]))) \\ for a prime p == 3 (mod 4), returns (p+1) * ord(5, p) / ord(2+-i, p)
isA385167(p) = isprime(p) && p%4==3 && quot(p)%2==0

Crossrefs

Cf. A002145, A385165 (list of ord(2+-i,p)), A385166 (list of (p+1) * ord(5,p) / ord(2+-i,p)).

Subsequence of the intersection of A122869 and A385168. Contains A385180 as a subsequence.

Sequence in context: A239437 A385168 A140542 * A101983 A139953 A111067

Adjacent sequences: A385164 A385165 A385166 * A385168 A385169 A385170

Keyword

nonn,easy

Author

Jianing Song, Jun 20 2025

Status

approved