OFFSET
1,7
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Şaban Alaca and Kenneth S. Williams, Evaluation of the convolution sums ..., Journal of Number Theory, Vol. 124, No. 2 (2007) pp. 491-510.
Mathieu Lemire and Kenneth S. Williams, Evaluation of two convolution sums involving the sum of divisors function, Bulletin of the Australian Mathematical Society, Vol. 73, No. 1 (2006), pp. 107-115.
Emmanuel Royer, Evaluating convolution sums of the divisor function by quasimodular forms, International Journal of Number Theory, Vol. 3, No. 2 (2007) pp. 231-261.
FORMULA
a(n) = Sum_{k < n/5} sigma(k) * sigma(n-5*k).
a(n) = 5*sigma_3(n)/312 + 125*sigma_3(n/5)/312 + (1/24 - n/20)*sigma(n) + (1/24 - n/4)*sigma(n/5) - c_5(n)/130, where sigma_3(n/5) = sigma(n/5) = 0 if n is not divisible by 5, and c_5(n) is the coefficient of q^n in the expansion of (eta(q) * eta(q^5))^4 (A030210).
MATHEMATICA
a[n_] := Sum[DivisorSigma[1, k] * DivisorSigma[1, n - 5*k], {k, 1, (n - 1)/5}]; Array[a, 100]
(* or *)
c[n_] := SeriesCoefficient[q * (QPochhammer[q] * QPochhammer[q^5])^4, {q, 0, n}]; a[n_] := 5 * DivisorSigma[3, n]/312 + If[Divisible[n, 5], 125 * DivisorSigma[3, n/5]/312, 0] - n * DivisorSigma[1, n]/20 - If[Divisible[n, 5], n * DivisorSigma[1, n/5]/4, 0] + DivisorSigma[1, n]/24 + If[Divisible[n, 5], DivisorSigma[1, n/5]/24, 0] - c[n]/130; Array[a, 100]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 09 2021
STATUS
approved