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 A385221 #19 Jun 28 2025 12:53:42 %S A385221 3,7,11,19,23,31,43,47,59,67,71,73,79,83,89,103,107,127,131,139,151, %T A385221 163,167,179,191,199,211,223,227,233,239,251,263,271,281,283,307,311, %U A385221 331,337,347,359,367,379,383,419,431,439,443,463,467,479,487,491,499,503,523,547,563 %N A385221 Primes p such that multiplicative order of 4 modulo p is odd. %C A385221 The multiplicative order of 4 modulo a(n) is A385227(n). %C A385221 Primes p such that neither ord(2,p) nor ord(-2,p) is divisible by 4, where ord(a,m) is the multiplicative order of a modulo m. (Note that we have either (a) ord(2,p) = ord(-2,p) and both are even; (b) ord(-2,p) = 2*ord(2,p), ord(2,p) is odd, ord(-2,p) == 2 (mod 4); or (c) ord(2,p) = 2*ord(-2,p), ord(-2,p) is odd, ord(2,p) == 2 (mod 4)). %C A385221 Contains all primes congruent to 3 modulo 4 (A002145). %C A385221 Conjecture: this sequence has density 7/12 among the primes (see A014663). %H A385221 Jianing Song, <a href="/A385221/b385221.txt">Table of n, a(n) for n = 1..10000</a> %t A385221 Select[Prime[Range[200]], OddQ[MultiplicativeOrder[4, #]] &] (* _Paolo Xausa_, Jun 28 2025 *) %o A385221 (PARI) isA385221(p) = isprime(p) && (p!=2) && znorder(Mod(4,p))%2 %Y A385221 Contains A002145, A014663, and A163183. %Y A385221 Cf. A385227 (the actual multiplicative orders). %Y A385221 Cf. other bases: A014663 (base 2), A385220 (base 3), this sequence (base 4), A385192 (base 5), A163183 (base -2), A385223 (base -3), A385224 (base -4), A385225 (base -5). %K A385221 nonn,easy %O A385221 1,1 %A A385221 _Jianing Song_, Jun 22 2025