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

8, 10, 12, 14, 15, 18, 20, 24, 26, 28, 32, 33, 34, 35, 38, 39, 40, 42, 44, 50, 51, 54, 55, 57, 58, 62, 63, 65, 66, 68, 74, 76, 77, 78, 80, 82, 85, 86, 87, 88, 91, 92, 94, 95, 96, 98, 99, 106, 110, 111, 112, 114, 115, 116, 117, 119, 122, 123, 124, 125, 129, 130, 132, 134, 136, 138, 140, 143, 146

Offset: 1

Author

Robert Israel, Aug 04 2025

Keywords

• nonn

Comments

Numbers k >= 2 that are not congruent to a square mod A001414(k).

Contains no primes or squares.

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.

Contains p*q if p and q are primes and p + q has a prime factor == 3 (mod 4) with multiplicity 1.

Examples

			a(3) = 12 is a term because A001414(12) = 2*2+3 = 7 and 12 is a quadratic nonresidue mod 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414.

Programs

Maple

sopfr:= proc(n) local t; add(t[1]*t[2], t=ifactors(n)[2]) end proc:
filter:= x -> numtheory:-quadres(x,sopfr(x))=-1:
select(filter, [$2..1000]);

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