A309129 Numbers n such that -n is a quadratic nonresidue modulo all odd primes p <= sqrt(n) which do not divide n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 33, 37, 40, 42, 43, 45, 48, 57, 58, 60, 63, 67, 70, 72, 78, 85, 88, 93, 100, 102, 105, 112, 120, 130, 133, 135, 147, 148, 163, 165, 168, 177, 190, 210, 232, 240, 247, 253, 267, 268, 273, 280, 312, 330, 333, 345, 357, 385, 408, 462, 520, 522, 652, 708, 760, 840, 928, 1320, 1365, 1467, 1848

Offset: 1

Author

Richard N. Smith, Jul 13 2019

Keywords

• nonn

Comments

Contains A000926 and A003173 (except the term 11) as subsequences.

Conjecture: 1848 is the last term of this sequence.

Examples

			42 is in this sequence because sqrt(42) = 6.480740..., and -42 is quadratic nonresidue mod all odd primes < 6.480740... not dividing 42 (only mod 5).
67 is in this sequence because sqrt(67) = 8.185352..., and -67 is quadratic nonresidue mod all odd primes < 8.185352... not dividing 67 (mod 3, mod 5 and mod 7).
17 is not in this sequence because -17 is quadratic residue mod 3 and 3 < sqrt(17) and 3 does not divide 17.
90 is not in this sequence because -90 is quadratic residue mod 7 and 7 < sqrt(90) and 7 does not divide 90.
For n < 9, the range of p is empty, thus the numbers n < 9 are trivially in this sequence.
		

Crossrefs

Cf. A000926, A003173.

Programs

PARI

a(n)=forprime(p=3, ,if(kronecker(-n,p)==1,return(p)))
for(k=1, 10^6,if(a(k)>sqrt(k),print1(k, ", ")))