

A000003


Number of classes of primitive positive definite binary quadratic forms of discriminant D = 4n; or equivalently the class number of the quadratic order of discriminant D = 4n.
(Formerly M0196 N0073)


49



1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 2, 2, 4, 2, 2, 4, 2, 3, 4, 4, 2, 3, 4, 2, 6, 3, 2, 6, 4, 3, 4, 4, 4, 6, 4, 2, 6, 4, 4, 8, 4, 3, 6, 4, 4, 5, 4, 4, 6, 6, 4, 6, 6, 4, 8, 4, 2, 9, 4, 6, 8, 4, 4, 8, 8, 3, 8, 8, 4, 7, 4, 4, 10, 6, 6, 8, 4, 5, 8, 6, 4, 9, 8, 4, 10, 6, 4, 12, 8, 6, 6, 4, 8, 8, 8, 4, 8, 6, 4
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OFFSET

1,5


COMMENTS

It seems that 2*a(n) gives the degree of the minimal polynomial of (k_n)^2 where k_n is the nth singular value, i.e., K(sqrt(1k_n^2))/K(k_n)==sqrt(n) (and K is the elliptic integral of the first kind: K(x) := hypergeom([1/2,1/2],[1], x^2)).
Also, when setting K3(x)=hypergeom([1/3,2/3],[1], x^3) and solving for x such that K3((1x^3)^(1/3))/K3(x)==sqrt(n), then the degree of the minimal polynomial of x^3 is every third term of this sequence, or so it seems. (End)
a(n) appears to be the degree of Klein's jinvariant j(sqrt(n)) as an algebraic integer.  Li Han, Mar 02 2020


REFERENCES

D. A. Buell, Binary Quadratic Forms. SpringerVerlag, NY, 1989, pages 20 and 231234.[Dics means D =  Discriminant (see p. 223), and only squarefree cases appear on pp. 231234, but not on p. 20.  Wolfdieter Lang, May 15 2021]
H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS



MATHEMATICA

a[1] = 1; a[n_] := (k0 = k /. FindRoot[EllipticK[1k^2]/EllipticK[k^2] == Sqrt[n], {k, 1/2, 10^10, 1}, WorkingPrecision > 600, MaxIterations > 100]; Exponent[ MinimalPolynomial[RootApproximant[k0^2, 24], x], x]/2); Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 1, 100}] (* JeanFrançois Alcover, Jul 21 2015, after Joerg Arndt *)


PROG

(Magma) O1 := MaximalOrder(QuadraticField(D)); _, f := IsSquare(D div Discriminant(O1)); ClassNumber(sub<O1f>);
(PARI) {a(n) = qfbclassno(4*n)}; /* Michael Somos, Jul 16 1999 */


CROSSREFS

See A014599 for discriminant (4n1).


KEYWORD

nonn,nice,easy


AUTHOR



STATUS

approved



