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

1,1

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.).

Links

Table of n, a(n) for n=1..76.

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.

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);

Crossrefs

Cf. A117728, A005875.

Sequence in context: A371910 A023820 A225355 * A172307 A108039 A367013

Adjacent sequences: A117723 A117724 A117725 * A117727 A117728 A117729

Keyword

nonn

Author

N. J. A. Sloane, Apr 14 2006

Status

approved