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

1,2

Comments

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

Conjecture: 1848 is the last term of this sequence.

Links

Table of n, a(n) for n=1..84.

Example

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.

Prog

(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, ", ")))

Crossrefs

Cf. A000926, A003173.

Sequence in context: A354581 A193096 A353744 * A307345 A033110 A049812

Adjacent sequences: A309126 A309127 A309128 * A309130 A309131 A309132

Keyword

nonn

Author

Richard N. Smith, Jul 13 2019

Status

approved