A006641 Class number of forms with discriminant -A003657(n), or equivalently class number of imaginary quadratic field with discriminant -A003657(n).

1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 3, 2, 4, 2, 1, 5, 2, 2, 4, 4, 3, 1, 4, 7, 5, 3, 4, 6, 2, 2, 8, 5, 6, 3, 8, 2, 6, 10, 4, 2, 5, 5, 4, 4, 3, 10, 2, 7, 6, 4, 10, 1, 8, 11, 4, 5, 8, 4, 2, 13, 4, 9, 4, 3, 6, 14, 4, 7, 5, 4, 12, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
• H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=1..3000
• S. R. Finch, Class number theory
• Steven R. Finch, Class number theory [Cached copy, with permission of the author]
• Rick L. Shepherd, Binary quadratic forms and genus theory, Master of Arts Thesis, University of North Carolina at Greensboro, 2013.
• Eric Weisstein's World of Mathematics, Class Number

Crossrefs

Cf. A003657.

Programs

Mathematica

FundamentalDiscriminantQ[n_Integer] := n != 1 && (Mod[n, 4] == 1 || !Unequal[ Mod[n, 16], 8, 12]) && SquareFreeQ[n/2^IntegerExponent[n, 2]] (* via Eric W. Weisstein *);
NumberFieldClassNumber@ Sqrt@ # & /@ Select[-Range@ 300, FundamentalDiscriminantQ]

PARI

for(n=1, 300, if(isfundamental(-n), print1(quadclassunit(-n).no, ", "))) \\ Andrew Howroyd, Jul 23 2018

Sage

[1] + [QuadraticField(-n, 'a').class_number() for n in (0..200) if is_fundamental_discriminant(-n) and not is_square(n)] # G. C. Greubel, Mar 01 2019