

A330165


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


0



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
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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), 5558 and 64.
P. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arith., 22 (1973), 117124.


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



