A384948 -id:A384948 - cp's OEIS frontend

A385168 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) > 1. Here ord(a,m) is the multiplicative order of a modulo m.

11, 79, 83, 131, 139, 191, 199, 211, 239, 251, 307, 331, 347, 359, 419, 439, 479, 491, 503, 571, 587, 599, 659, 691, 719, 811, 839, 863, 919, 947, 971, 1019, 1039, 1051, 1091, 1103, 1171, 1223, 1231, 1279, 1291, 1319, 1399, 1439, 1451, 1499, 1523, 1531, 1559, 1567, 1571, 1619, 1667, 1759

Offset: 1

Jianing Song, Jun 20 2025

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) > 1.
The smallest terms congruent to 1 or 4 modulo 5 that are not in A385167 are 139, 191, 419, 659, ...
The smallest terms congruent to 2 or 3 modulo 5 that are not in A384948 are 5683, 6287, 9463, 9923, ...

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

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

A384948 and A385167 (which contains A385180) are subsequences.

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)
isA385168(p) = isprime(p) && p%4==3 && quot(p) > 1

139 is a term since the multiplicative order of 2+-i modulo 139 is 1932, and (140*ord(5,139))/1932 = 5 > 1.

5683 is a term since the multiplicative order of 2+-i modulo 5683 is 1537928, and (5684*ord(5,5683))/1537928 = 7 > 1.

cp's OEIS Frontend

A385168 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) > 1. Here ord(a,m) is the multiplicative order of a modulo m.

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