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A117726 Moreno and Wagstaff's arithmetical function T(n). 2

%I

%S 2,4,4,4,8,8,4,8,10,8,12,8,8,16,8,8,16,12,12,16,16,8,12,16,10,24,16,8,

%T 24,16,12,16,16,16,24,20,8,24,16,16,32,16,12,24,24,16,20,16,18,28,24,

%U 16,24,32,16,32,16,8,36,16,24,32,20,16,32,32,12,32,32,16,28,24,16,40,28,24

%N Moreno and Wagstaff's arithmetical function T(n).

%C Also 4 times Kronecker's function F(n).

%C F(n) is the number of odd classes of binary quadratic forms ax^2+2bxy+cy^2 of discriminant b^2-ac = -n, where classes of the shape a(x^2+y^2) are counted as 1/2 and "odd" means that at least one of a and c is odd.

%D L. Kronecker, Crelle, Vol. LVII (1860), p. 248; Werke, Vol. IV, p. 188.

%D C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 43.

%D H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see pp. 323 (definition of F), 338 (g.f.).

%F G.f. for F(n): Sum_{n >= 1} F(n) q^n = (q^(1/4) / Sum_{ m=-infinity, infinity } q^( (2*m+1)^2/4 )) * Sum{ n=-infinity, infinity } q^(n^2+n-1)/(1-q^(2*n-1))^2.

%p t10:=add( q^( (2*m+1)^2/4 ),m=-20..20); t1:=series(q^(1/4)/t10,q,100); t2:=add( q^(n^2+n-1)/(1-q^(2*n-1))^2,n=1..100): series(4*t1*t2,q,100);

%Y Cf. A117728, A005875.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Apr 14 2006

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Last modified February 3 12:24 EST 2023. Contains 360035 sequences. (Running on oeis4.)