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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.

%I A000003 M0196 N0073 #82 Aug 14 2022 17:00:40
%S A000003 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,
%T A000003 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,
%U A000003 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
%N 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.
%C A000003 From _Joerg Arndt_, Sep 02 2008: (Start)
%C A000003 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)).
%C A000003 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)
%C A000003 a(n) appears to be the degree of Klein's j-invariant j(sqrt(-n)) as an algebraic integer. - _Li Han_, Mar 02 2020
%D A000003 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]
%D A000003 H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 514.
%D A000003 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000003 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000003 N. J. A. Sloane, <a href="/A000003/b000003.txt">Table of n, a(n) for n = 1..20000</a>
%H A000003 Harriet Fell, Morris Newman, Edward Ordman, <a href="https://dx.doi.org/10.6028/jres.067B.006">Tables of genera of groups of linear fractional transformations</a>, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
%H A000003 Daniel Shanks, <a href="https://doi.org/10.1090/S0025-5718-1960-0120203-6">On the Conjecture of Hardy & Littlewood concerning the Number of Primes of the Form n^2 + a</a>, Math. Comp. 14 (1960), 320-332. (Table 1 gives first 100 terms.)
%H A000003 D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>. Math. Comp. 21 (1967) 689-694.
%H A000003 D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
%t A000003 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_ *)
%o A000003 (Magma) O1 := MaximalOrder(QuadraticField(D)); _,f := IsSquare(D div Discriminant(O1)); ClassNumber(sub<O1|f>);
%o A000003 (PARI) {a(n) = qfbclassno(-4*n)}; /* _Michael Somos_, Jul 16 1999 */
%Y A000003 See A014599 for discriminant -(4n-1).
%Y A000003 A006643 is a subsequence.
%K A000003 nonn,nice,easy
%O A000003 1,5
%A A000003 _N. J. A. Sloane_