A003640 Number of genera of imaginary quadratic field with discriminant -k, k = A003657(n). (Formerly M0090)

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 4, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 4, 2, 1, 1, 4, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 1, 2, 2, 2, 1, 1, 4, 4, 2, 2, 1, 2, 2, 2, 1, 4, 2, 4, 1, 4, 2, 1, 4, 4, 1, 2, 2, 2, 2, 2, 2, 2, 1, 4, 1, 4, 2, 2, 2, 2, 1, 2

Offset

1,6

Comments

The number of genera of a quadratic field is equal to the number of elements x in the class group such that x^2 = e where e is the identity. - Jianing Song, Jul 24 2018

References

D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

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

Index entries for sequences related to quadratic fields

Formula

a(n) = 2^(t-1) where t = number of distinct prime divisors of A003657(n).

a(n) = 2^(omega(A003657(n)) - 1).

Mathematica

okQ[n_] := n != 1 && SquareFreeQ[n/2^IntegerExponent[n, 2]] && (Mod[n, 4] == 3 || Mod[n, 16] == 8 || Mod[n, 16] == 4);
Reap[For[n = 1, n <= 1000, n++, If[okQ[n], Sow[2^(PrimeNu[n]-1)]]]][[2, 1]] (* Jean-François Alcover, Aug 16 2019, after Andrew Howroyd *)

Prog

(PARI) for(n=1, 1000, if(isfundamental(-n), print1(2^(omega(n) - 1), ", "))) \\ Andrew Howroyd, Jul 24 2018
(PARI) for(n=1, 1000, if(isfundamental(-n), print1(2^#select(t->t%2==0, quadclassunit(-n).cyc), ", "))) \\ Andrew Howroyd, Jul 24 2018

Crossrefs

Cf. A001221 (omega), A003641, A003642, A003643, A003657, A191408.

Sequence in context: A305392 A175301 A214074 * A359944 A359308 A107459

Adjacent sequences: A003637 A003638 A003639 * A003641 A003642 A003643

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Mira Bernstein

Extensions

Name clarified and offset corrected by Jianing Song, Jul 24 2018

Status

approved