A317987 Discriminants of orders of imaginary quadratic fields with 2 classes per genus, negated.

39, 55, 56, 63, 68, 80, 128, 136, 144, 155, 156, 171, 184, 196, 203, 208, 219, 220, 224, 252, 256, 259, 260, 264, 275, 276, 291, 292, 308, 320, 323, 328, 336, 355, 360, 363, 384, 387, 388, 400, 456, 468, 475, 504, 507, 528, 544, 552, 564, 568, 576, 580, 592, 600, 603, 612, 616, 624, 640

Offset: 1

Author

Jianing Song, Oct 02 2018

Keywords

• nonn
• fini

Comments

k is a term iff the form class group of positive binary quadratic forms with discriminant -k is isomorphic to (C_2)^r X C_4.

This is a subsequence of A133676, so it's finite. It seems that this sequence has 324 terms, the largest being 87360.

The smallest number in A133676 but not here is 3600.

Links

• Jianing Song, Table of n, a(n) for n = 1..324
• Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Crossrefs

Cf. A003171, A133676.

Fundamental terms are listed in A319983.

Programs

PARI

isA317987(n) = (-n)%4 < 2 && 2^(1+#quadclassunit(-n)[2])==quadclassunit(-n)[1]

Formula

The form class groups of positive binary quadratic forms with discriminant -39, -55, -56, -63, -68, -80 and -128 are all isomorphic to C_4, so 39, 55, 56, 63, 68, 80 and 128 are all members of this sequence.