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A000003
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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)
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10
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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
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1,5
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COMMENTS
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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)
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REFERENCES
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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).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..5000
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PROG
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(MAGMA) O1 := MaximalOrder(QuadraticField(D)); _, f := IsSquare(D div Discriminant(O1)); ClassNumber(sub<O1|f>);
(PARI) a(n)=qfbclassno(-4*n)
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CROSSREFS
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Sequence in context: A029405 A029350 A166597 * A168208 A197081 A029395
Adjacent sequences: A000001 A000002 * A000004 A000005 A000006 A000007
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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