A317987 -id:A317987 - cp's OEIS frontend

A133676 Negative discriminants with form class group of exponent 4 (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

Offset: 1

David Brink, Dec 29 2007

The sequence is finite. It appears to have exactly 485 terms, the largest being 887040.
The finiteness of the sequence was proved by Earnest and Estes.
I found the 485 terms with PARI and didn't find any other up to 50000000.

David Brink, Table of n, a(n) for n = 1..485
David Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms (Video abstract)
David Brink, Five peculiar theorems on simultaneous representation of primes by quadratic forms, J. Number Theory 129 (2009), no. 2, 464-468.
A. G. Earnest and D. R. Estes, An algebraic approach to the growth of class numbers of binary quadratic lattices, Mathematika 28 (1981), no. 2, 160--168.
Journal of Number Theory, Video Abstracts

Cf. A003173, A317987 (subsequence).

a(n) = if(n%4==0 || n%4==3, my(v = quadclassunit(-n)[2]); (#v > 0) && (v[1] == 4), 0) \\ Jianing Song, Sep 24 2022

A330219 Negative discriminants with form class group isomorphic to C_4 (negated).

Original entry on oeis.org

39, 55, 56, 63, 68, 80, 128, 136, 144, 155, 156, 171, 184, 196, 203, 208, 219, 220, 252, 256, 259, 275, 291, 292, 323, 328, 355, 363, 387, 388, 400, 475, 507, 568, 592, 603, 667, 723, 763, 772, 955, 1003, 1027, 1227, 1243, 1387, 1411, 1467, 1507, 1555

Offset: 1

Jianing Song, Dec 05 2019

It seems that this is the full list.

Equivalently, negative discriminants of orders whose class group is isomorphic to C_4 (negated). - Jianing Song, May 17 2021

Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013

Cf. A133675 (negative discriminants with form class group isomorphic to the trivial group), A322710 (isomorphic to C_2), A328825 (isomorphic to C_3), A329182 (isomorphic to C_2 X C_2), this sequence (isomorphic to C_4).

Subsequence of A133676 and A317987. Cf. also A013658.

isA330219(d) = (d>0) && ((d%4==0)||(d%4==3)) && quadclassunit(-d)[2]==[4] \\ Jianing Song, May 17 2021

A319983 Discriminants of imaginary quadratic fields with 2 classes per genus, negated.

Original entry on oeis.org

39, 55, 56, 68, 136, 155, 184, 203, 219, 259, 260, 264, 276, 291, 292, 308, 323, 328, 355, 388, 456, 552, 564, 568, 580, 616, 651, 667, 723, 763, 772, 820, 852, 868, 915, 952, 955, 987, 1003, 1027, 1032, 1060, 1128, 1131, 1140, 1204, 1227, 1240, 1243, 1288, 1387, 1411, 1443

Offset: 1

Jianing Song, Oct 02 2018

Fundamental terms of A317987.
k is a term iff the class group of Q[sqrt(-k)], or 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 161 terms, the largest being 40755.

			See examples in A317987.

Jianing Song, Table of n, a(n) for n = 1..161
Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.

Cf. A003644, A133676.

Subsequence of A317987.

isA319983(n) = isfundamental(-n) && 2^(1+#quadclassunit(-n)[2])==quadclassunit(-n)[1]

cp's OEIS Frontend

A133676 Negative discriminants with form class group of exponent 4 (negated).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A330219 Negative discriminants with form class group isomorphic to C_4 (negated).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A319983 Discriminants of imaginary quadratic fields with 2 classes per genus, negated.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI