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

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

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy
• nice

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

Crossrefs

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

Programs

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 *)

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

Formula

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

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

Extensions

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