

A227735


Negative fundamental discriminants with cyclic class groups of composite order (negated).


1



39, 55, 56, 68, 87, 95, 104, 111, 116, 119, 136, 143, 152, 155, 159, 164, 183, 184, 199, 203, 212, 215, 219, 239, 244, 247, 248, 259, 287, 291, 292, 295, 296, 299, 303, 319, 323, 327, 328, 335, 339, 344, 355, 356, 367, 371, 376, 388, 391, 395, 404, 407, 411
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OFFSET

1,1


COMMENTS

Absolute values of fundamental discriminants of imaginary quadratic fields whose class groups are cyclic of composite order. Of course every class group of squarefree order is necessarily cyclic. (This means the negatives of negative fundamental discriminants with class groups of composite squarefree orders are a proper subsequence.)
The nth line of the linked file gives the order of the class group (the class number) corresponding to the fundamental discriminant a(n).
The negative of each term is either a negative fundamental discriminant or the product of exactly one positive prime discriminant and one negative prime discriminant where the product contains at most one factor in {8, 4, 8} and is unique disregarding order. In the first case, the class number is odd (because the 2rank of the class group is 0); in the second, even (because the 2rank is 1).


LINKS

Rick L. Shepherd, Table of n, a(n) for n = 1..10000
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
Rick L. Shepherd, Orders of corresponding class groups
Index entries for sequences related to quadratic fields


EXAMPLE

The fundamental discriminant 39 = (3)(13) has a cyclic class group of order 4, which is composite (but not squarefree). The fundamental discriminant 104 = (8)(13) has a cyclic class group of order 6, which is composite. The fundamental discriminant 239 is itself a prime discriminant with cyclic class group of order 15, also composite (but not divisible by 2).


PROG

(PARI)
{default(realprecision, 100);
terms_wanted = 10000;
t = 0; k = 0;
while(t < terms_wanted,
k++;
if(isfundamental(k),
F = bnfinit(quadpoly(k, x), , [6, 6, 4]);
if(bnfcertify(F) <> 1,
print("Certify failed for ", k, "  exiting (",
t, " terms found)"); break);
if(length(F.clgp.cyc) == 1 &&
isprime(F.clgp.cyc[1]) == 0,
t++;
write("b227735.txt", t, " ", k);
write("a227735.txt", t, " ", F.clgp.cyc[1]))))}


CROSSREFS

Cf. A227734.
Sequence in context: A317987 A330219 A013658 * A319983 A165346 A268083
Adjacent sequences: A227732 A227733 A227734 * A227736 A227737 A227738


KEYWORD

nonn


AUTHOR

Rick L. Shepherd, Jul 29 2013


STATUS

approved



