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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 (list; graph; refs; listen; history; text; internal format)
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.

G√ľnther 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

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Last modified August 4 09:03 EDT 2020. Contains 336201 sequences. (Running on oeis4.)