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A346193
Convolution of level 5 of the divisor function.
1
0, 0, 0, 0, 0, 1, 3, 4, 7, 6, 15, 17, 27, 34, 36, 52, 64, 75, 91, 102, 122, 155, 169, 193, 228, 263, 276, 326, 349, 415, 430, 500, 520, 620, 681, 727, 741, 881, 880, 1090, 1020, 1192, 1178, 1375, 1513, 1590, 1557, 1809, 1756, 2274, 2024, 2323, 2245, 2626, 2865
OFFSET
1,7
LINKS
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]
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 09 2021
STATUS
approved