A000926 -id:A000926 - cp's OEIS frontend

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

Richard N. Smith, Jul 13 2019

nonn

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

Conjecture: 1848 is the last term of this sequence.

			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.

Cf. A000926, A003173.

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

A330165 Odd terms in A003171: negated odd discriminants of orders of imaginary quadratic fields with 1 class per genus.

Original entry on oeis.org

3, 7, 11, 15, 19, 27, 35, 43, 51, 67, 75, 91, 99, 115, 123, 147, 163, 187, 195, 235, 267, 315, 403, 427, 435, 483, 555, 595, 627, 715, 795, 1155, 1435, 1995, 3003, 3315

Offset: 1

Jianing Song, Dec 04 2019

A003171 = 4*A000926 U {a(n)}.
Note that d is in A000926 (i.e., 4d is in A003171) if and only if: for all gcd(d,k) = 1, if k^2 < 3d, then d + k^2 is either a prime, or twice a prime, or the square of a prime, or 8 or 16. It seems that d is in this sequence if and only if: for all odd k, gcd(d,k) = 1, if k^2 < 3d, then (d + k^2)/4 is either a prime or the square of a prime.
It is conjectured that this is the full list. Otherwise, there could be at most one more term d such that -d is a fundamental discriminant.

			For d = 315, (d + k^2)/4 can be 79, 109, 121, 151, 169, 211, 289, each is a prime or the square of a prime.
For d = 3315 which is the largest known odd term in A003171, (d + k^2)/4 can be: 829, 841, 859, 919, 961, 1039, 1069, 1171, 1249, 1291, 1381, 1429, 1531, 1699, 1759, 1951, 2089, 2161, 2311, 2389, 2551, 2809, 3181, each is a prime or the square of a prime.

Günther Frei, Euler's convenient numbers, Math. Intell. Vol. 7 No. 3 (1985), 55-58 and 64.
P. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith., 22 (1973), 117-124.

Cf. A003171, A000926.

isA330165(n) = (n>0) && (n%4==3) && !#select(k->k<>2, quadclassunit(-n).cyc)

cp's OEIS Frontend

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

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