A003641 Number of genera of imaginary quadratic field with discriminant -k, k = A039957(n). (Formerly M0061)

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

1,4

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

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

Table of n, a(n) for n=1..105.

Index entries for sequences related to quadratic fields

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A353376 A279848 A001826 * A355241 A165190 A025890

Adjacent sequences: A003638 A003639 A003640 * A003642 A003643 A003644

Keyword

nonn

Author

N. J. A. Sloane, Mira Bernstein

Extensions

Name clarified by Jianing Song, Jul 24 2018

Status

approved