This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385168 #20 Jun 23 2025 14:41:40 %S A385168 11,79,83,131,139,191,199,211,239,251,307,331,347,359,419,439,479,491, %T A385168 503,571,587,599,659,691,719,811,839,863,919,947,971,1019,1039,1051, %U A385168 1091,1103,1171,1223,1231,1279,1291,1319,1399,1439,1451,1499,1523,1531,1559,1567,1571,1619,1667,1759 %N 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. %C A385168 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 A385168 List of p = A002145(k) such that A385166(k) > 1. %C A385168 The smallest terms congruent to 1 or 4 modulo 5 that are not in A385167 are 139, 191, 419, 659, ... %C A385168 The smallest terms congruent to 2 or 3 modulo 5 that are not in A384948 are 5683, 6287, 9463, 9923, ... %H A385168 Jianing Song, <a href="/A385168/b385168.txt">Table of n, a(n) for n = 1..10001</a> %F A385168 139 is a term since the multiplicative order of 2+-i modulo 139 is 1932, and (140*ord(5,139))/1932 = 5 > 1. %F A385168 5683 is a term since the multiplicative order of 2+-i modulo 5683 is 1537928, and (5684*ord(5,5683))/1537928 = 7 > 1. %o A385168 (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 A385168 isA385168(p) = isprime(p) && p%4==3 && quot(p) > 1 %Y A385168 Cf. A002145, A385165 (list of ord(2+-i,p)), A385166 (list of (p+1) * ord(5,p) / ord(2+-i,p)). %Y A385168 A384948 and A385167 (which contains A385180) are subsequences. %K A385168 nonn,easy %O A385168 1,1 %A A385168 _Jianing Song_, Jun 20 2025