cp's OEIS Frontend

A385679 Numbers k that are not congruent to a square mod sopfr(k).

%I A385679 #16 Aug 11 2025 05:18:10
%S A385679 8,10,12,14,15,18,20,24,26,28,32,33,34,35,38,39,40,42,44,50,51,54,55,
%T A385679 57,58,62,63,65,66,68,74,76,77,78,80,82,85,86,87,88,91,92,94,95,96,98,
%U A385679 99,106,110,111,112,114,115,116,117,119,122,123,124,125,129,130,132,134,136,138,140,143,146
%N A385679 Numbers k that are not congruent to a square mod sopfr(k).
%C A385679 Numbers k >= 2 that are not congruent to a square mod A001414(k).
%C A385679 Contains no primes or squares.
%C A385679 Contains p^k if p is prime and k is an odd number with a prime factor q such that p is a quadratic nonresidue mod q.
%C A385679 Contains p*q if p and q are primes and p + q has a prime factor == 3 (mod 4) with multiplicity 1.
%H A385679 Robert Israel, <a href="/A385679/b385679.txt">Table of n, a(n) for n = 1..10000</a>
%e A385679 a(3) = 12 is a term because A001414(12) = 2*2+3 = 7 and 12 is a quadratic nonresidue mod 7.
%p A385679 sopfr:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
%p A385679 filter:= x -> numtheory:-quadres(x,sopfr(x))=-1:
%p A385679 select(filter, [$2..1000]);
%o A385679 (PARI) isok(k) = if (k>1, my(f=factor(k)); !issquare(Mod(k, sum(i=1, #f~, f[i, 1]*f[i, 2])))); \\ _Michel Marcus_, Aug 04 2025
%Y A385679 Cf. A001414.
%K A385679 nonn
%O A385679 1,1
%A A385679 _Robert Israel_, Aug 04 2025