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A117726 Moreno and Wagstaff's arithmetical function T(n). 2
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 (list; graph; refs; listen; history; text; internal format)
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: A327333 A023820 A225355 * A172307 A108039 A103228

Adjacent sequences: A117723 A117724 A117725 * A117727 A117728 A117729

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Apr 14 2006

STATUS

approved

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Last modified December 9 04:13 EST 2022. Contains 358698 sequences. (Running on oeis4.)