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

%I A330165 #10 Dec 08 2019 12:31:49
%S A330165 3,7,11,15,19,27,35,43,51,67,75,91,99,115,123,147,163,187,195,235,267,
%T A330165 315,403,427,435,483,555,595,627,715,795,1155,1435,1995,3003,3315
%N A330165 Odd terms in A003171: negated odd discriminants of orders of imaginary quadratic fields with 1 class per genus.
%C A330165 A003171 = 4*A000926 U {a(n)}.
%C A330165 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.
%C A330165 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.
%H A330165 Günther Frei, <a href="https://doi.org/10.1007/BF03025809">Euler's convenient numbers</a>, Math. Intell. Vol. 7 No. 3 (1985), 55-58 and 64.
%H A330165 P. Weinberger, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa22/aa2221.pdf">Exponents of the class groups of complex quadratic fields</a>, Acta Arith., 22 (1973), 117-124.
%e A330165 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.
%e A330165 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.
%o A330165 (PARI) isA330165(n) = (n>0) && (n%4==3) && !#select(k->k<>2, quadclassunit(-n).cyc)
%Y A330165 Cf. A003171, A000926.
%K A330165 nonn,fini,more
%O A330165 1,1
%A A330165 _Jianing Song_, Dec 04 2019