

A000003


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


15



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

From Joerg Arndt, Sep 02 2008: (Start)
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)


REFERENCES

H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations. J. Res. Nat. Bur. Standards Sect. B 67B 1963 6168.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689694; 22 (1968), 699.
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

N. J. A. Sloane, Table of n, a(n) for n = 1..5000


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

Sequence in context: A198260 A029350 A166597 * A234398 A168208 A197081
Adjacent sequences: A000001 A000002 * A000004 A000005 A000006 A000007


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane.


STATUS

approved



