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

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..36.

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.

Example

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.

Prog

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

Crossrefs

Cf. A003171, A000926.

Sequence in context: A220494 A194440 A220520 * A228436 A189364 A022797

Adjacent sequences: A330162 A330163 A330164 * A330166 A330167 A330168

Keyword

nonn,fini,more

Author

Jianing Song, Dec 04 2019

Status

approved