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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 (list; graph; refs; listen; history; text; internal format)
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 n-th singular value, i.e. K(sqrt(1-k_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((1-x^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 61-68.

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 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<O1|f>);

(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

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Last modified April 18 23:08 EDT 2014. Contains 240734 sequences.