A008457 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^3.

1, 7, 28, 71, 126, 196, 344, 583, 757, 882, 1332, 1988, 2198, 2408, 3528, 4679, 4914, 5299, 6860, 8946, 9632, 9324, 12168, 16324, 15751, 15386, 20440, 24424, 24390, 24696, 29792, 37447, 37296, 34398, 43344, 53747, 50654, 48020, 61544, 73458

Offset

1,2

Comments

The modular form (e(1)-e(2))(e(1)-e(3)) for GAMMA_0 (2) (with constant term -1/16 omitted).

a(n) = r_8(n)/16, where r_8(n) = A000143(n) is the number of integral solutions of Sum_{j=1..8} x_j^2 = n (with the order of the summands respected). See the Grosswald reference, and the Hardy reference, pp. 146-147, eq. (9.9.3) and sect. 9.10. - Wolfdieter Lang, Jan 09 2017

References

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.6).

Emil Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121, eq. (9.19).

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

F. Hirzebruch, T. Berger and R. Jung, Manifolds and Modular Forms, Vieweg, 1994, pp. 77, 133.

Hans Petersson, Modulfunktionen und Quadratische Formen, Springer-Verlag, 1982; p. 179.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Meinhard Peters, Sums of nine squares, Acta Arith., Vol. 102 (2002), pp. 131-135.

Formula

Multiplicative with a(2^e) = (8^(e+1)-15)/7, a(p^e) = (p^(3*e+3)-1)/(p^3-1), p > 2. - Vladeta Jovovic, Sep 10 2001

a(n) = (-1)^n*(sum of cubes of even divisors of n - sum of cubes of odd divisors of n), see A051000. Sum_{n>0} n^3*x^n*(15*x^n-(-1)^n)/(1-x^(2*n)). - Vladeta Jovovic, Oct 24 2002

G.f.: Sum_{k>0} k^3 x^k/(1 - (-x)^k). - Michael Somos, Sep 25 2005

G.f.: (1/16)*(-1+(Product_{k>0} (1-(-q)^k)/(1+(-q)^k))^8). [corrected by Vaclav Kotesovec, Sep 26 2015]

Dirichlet g.f. zeta(s)*zeta(s-3)*(1-2^(1-s)+2^(4-2s)), Dirichlet convolution of A001158 and the quasi-finite (1,-2,0,16,0,0,...). - R. J. Mathar, Mar 04 2011

A138503(n) = -(-1)^n * a(n).

Bisection: a(2*k-1) = A001158(2*k-1), a(2*k) = 8*A001158(k) - A051000(k), k >= 1. In the Hardy reference a(n) = sigma^*_3(n). - Wolfdieter Lang, Jan 07 2017

G.f.: (theta_3(x)^8 - 1)/16, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018

Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 384. - Vaclav Kotesovec, Sep 21 2020

Example

G.f. = q + 7*q^2 + 28*q^3 + 71*q^4 + 126*q^5 + 196*q^6 + 344*q^7 + 583*q^8 + ...

Maple

(1/16)*product((1+q^n)^8/(1-q^n)^8, n=1..60);

Mathematica

nmax = 40; Rest[CoefficientList[Series[Product[((1-(-q)^k)/(1+(-q)^k))^8, {k, 1, nmax}]/16, {q, 0, nmax}], q]] (* Vaclav Kotesovec, Sep 26 2015 *)
a[n_] := DivisorSum[n, (-1)^(n-#)*#^3&]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^8 - 1) / 16, {x, 0, n}]; (* Michael Somos, Aug 10 2018 *)
f[2, e_] := (8^(e+1)-15)/7; f[p_, e_] := (p^(3*e+3)-1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

Prog

(PARI) {a(n) = if( n<1, 0, (-1)^n * sumdiv(n, d, (-1)^d * d^3))}; /* Michael Somos, Sep 25 2005 */
(Python)
from math import prod
from sympy import factorint
def A008457(n): return prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 21 2024

Crossrefs

Cf. A000143, A001158, A051000, A064027, A002129, A048272, A138503.

Sequence in context: A033582 A176362 A358999 * A138503 A223765 A064951

Adjacent sequences: A008454 A008455 A008456 * A008458 A008459 A008460

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Status

approved