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%I A000003 M0196 N0073
%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 binary forms of discriminant D = -4n; 
               or equivalently class number of quadratic order of discriminant D 
               = -4n.
%C A000003 Comment from Joerg Arndt (arndt(AT)jjj.de), Sep 02 2008: (Start) It seems 
               that 2a(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 function 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)
%D A000003 H. Cohen, Course in Computational Alg. No. Theory, Springer, 1993, p. 
               514.
%D A000003 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 A000003 D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 
               689-694; 22 (1968), 699.
%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="b000003.txt">Table of n, a(n) for n = 1..5000</
               a>
%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)
%Y A000003 Sequence in context: A029405 A029350 A166597 this_sequence A029395 A029282 
               A029286
%Y A000003 Adjacent sequences: A000001 A000002 this_sequence A000004 A000005 A000006
%K A000003 nonn,nice,easy
%O A000003 1,5
%A A000003 N. J. A. Sloane (njas(AT)research.att.com).

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