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Discriminants of orders of imaginary quadratic fields with 2 classes per genus, negated.
4

%I #17 Oct 09 2018 01:48:09

%S 39,55,56,63,68,80,128,136,144,155,156,171,184,196,203,208,219,220,

%T 224,252,256,259,260,264,275,276,291,292,308,320,323,328,336,355,360,

%U 363,384,387,388,400,456,468,475,504,507,528,544,552,564,568,576,580,592,600,603,612,616,624,640

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

%C 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.

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

%C The smallest number in A133676 but not here is 3600.

%H Jianing Song, <a href="/A317987/b317987.txt">Table of n, a(n) for n = 1..324</a>

%H Rick L. Shepherd, <a href="http://libres.uncg.edu/ir/uncg/f/Shepherd_uncg_0154M_11099.pdf">Binary quadratic forms and genus theory</a>, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

%F 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.

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

%Y Cf. A003171, A133676.

%Y Fundamental terms are listed in A319983.

%K nonn,fini

%O 1,1

%A _Jianing Song_, Oct 02 2018