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)
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]
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 *)
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
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved