A003641 Number of genera of imaginary quadratic field with discriminant -k, k = A039957(n).

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

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn

Comments

In other words, this is the number of genera of those imaginary quadratic fields that have a discriminant which is odd and fundamental. The discriminant will be squarefree and of the form -4n+1. - Andrew Howroyd, Jul 25 2018

Examples

			a(4) = 2 because -15 = -A039957(4) and the number of genera of the quadratic field with discriminant -15 is 2. - _Andrew Howroyd_, Jul 25 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

• Index entries for sequences related to quadratic fields

Crossrefs

Cf. A001221 (omega), A003640, A003642, A039957.

Programs

Mathematica

2^(PrimeNu[Select[Range[1000], Mod[#, 4] == 3 && SquareFreeQ[#]&]] - 1) (* Jean-François Alcover, Jul 25 2019, after Andrew Howroyd *)

PARI

for(n=1, 1000, if(n%4==3 && issquarefree(n), print1(2^(omega(n) - 1), ", "))) \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = 2^(omega(A039957(n)) - 1). - Jianing Song, Jul 24 2018

Extensions

Name clarified by Jianing Song, Jul 24 2018