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

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

Offset: 1

Author

N. J. A. Sloane, Apr 14 2006

Keywords

• nonn

Comments

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

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.

References

• L. Kronecker, Crelle, Vol. LVII (1860), p. 248; Werke, Vol. IV, p. 188.
• C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 43.
• 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.).

Crossrefs

Cf. A117728, A005875.

Programs

Maple

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);

Formula

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.