

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)


45



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)
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 19 and 231234.
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

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Harriet Fell, Morris Newman, Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 6168.
Daniel Shanks, On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n^2 + a, Math. Comp. 14 (1960), 320332. (Table 1 gives first 100 terms.)
D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689694.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689694; 22 (1968), 699. [Annotated scanned copy]


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).
A006643 is a subsequence.
Sequence in context: A166597 A302490 A316841 * A234398 A168208 A333003
Adjacent sequences: A000001 A000002 * A000004 A000005 A000006 A000007


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



