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.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

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)

a(n) appears to be the degree of Klein's j-invariant j(sqrt(-n)) as an algebraic integer. - Li Han, Mar 02 2020

References

• D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pages 20 and 231-234.[Dics means D = - Discriminant (see p. 223), and only squarefree cases appear on pp. 231-234, 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

• 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 61-68.
• Daniel Shanks, On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n^2 + a, Math. Comp. 14 (1960), 320-332. (Table 1 gives first 100 terms.)
• D. Shanks, Generalized Euler and class numbers. Math. Comp. 21 (1967) 689-694.
• D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]

Crossrefs

See A014599 for discriminant -(4n-1).

A006643 is a subsequence.

Programs

Magma

O1 := MaximalOrder(QuadraticField(D)); _,f := IsSquare(D div Discriminant(O1)); ClassNumber(sub);

Mathematica

a[1] = 1; a[n_] := (k0 = k /. FindRoot[EllipticK[1-k^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}] (* Jean-François Alcover, Jul 21 2015, after Joerg Arndt *)

PARI

{a(n) = qfbclassno(-4*n)}; /* Michael Somos, Jul 16 1999 */