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A385180 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) is divisible by 4. Here ord(a,m) is the multiplicative order of a modulo m.

%I A385180 #25 Jun 23 2025 14:41:26
%S A385180 331,571,599,691,839,919,971,1039,1051,1171,1279,1291,1319,1399,1439,
%T A385180 1451,1571,1759,1879,2131,2411,2879,2971,3079,3251,3331,3491,3571,
%U A385180 3691,3851,4051,4079,4091,4211,4519,4639,4651,4679,4691,4759,4919,4931,5051,5119,5171,5279,5479,5519,5531
%N A385180 Primes p == 3 (mod 4) such that (p+1) * ord(5,p) / ord(2+-i,p) is divisible by 4. Here ord(a,m) is the multiplicative order of a modulo m.
%C A385180 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).
%C A385180 List of p = A002145(k) such that A385166(k) is divisible by 4.
%C A385180 Since in this case d(p) divides (p^2-1)/2, 5 must be a quadratic residue modulo p (see A385165).
%C A385180 By definition, a term that is in neither A385169 nor A385179 must be congruent to 31 or 79 modulo 80. The smallest such term is p = 1759 (ord(2+-i,p) = ((p+1)/4) * ord(5,p) = 128920); even if 1039 == 79 (mod 80), we have ord(2+-i,p) =  ((p+1)/8) * ord(5,p) = 22490 == 2 (mod 4), which means that 1039 is in A385179.
%H A385180 Jianing Song, <a href="/A385180/b385180.txt">Table of n, a(n) for n = 1..10000</a>
%e A385180 571 is a term since the multiplicative order of 2+-i modulo 571 is 40755, and (572*ord(5,571))/40755 = 4 is divisible by 4.
%o A385180 (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)
%o A385180 isA385180(p) = isprime(p) && p%4==3 && quot(p)%4==0
%Y A385180 Cf. A002145, A385165 (list of ord(2+-i,p)), A385166 (list of (p+1) * ord(5,p) / ord(2+-i,p)).
%Y A385180 Subsequence of A385167, which is itself a subsequence of intersection of A122869 and A385168.
%K A385180 nonn,easy
%O A385180 1,1
%A A385180 _Jianing Song_, Jun 20 2025