A218276 Convolution of level 2 of the divisor function.

0, 0, 1, 3, 7, 16, 22, 45, 49, 100, 95, 178, 161, 304, 250, 465, 372, 676, 525, 952, 720, 1280, 946, 1702, 1217, 2156, 1570, 2764, 1925, 3376, 2360, 4185, 2912, 4944, 3404, 6121, 4047, 6960, 4858, 8344, 5530, 9600, 6391, 11246, 7513, 12496, 8372, 14926, 9486

Offset: 1

Author

Michel Marcus, Oct 25 2012

Keywords

• nonn

Comments

Belongs to the family of convolution sums: Sum_{m < n*N} sigma(n)*sigma(n - N*m).

Named W2(n) by S. Alaca and K. S. Williams.

The convolution sum: Sum_{m < n} sigma(n)*sigma(n - m) = W1(n) is A000385(n+1).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• S. Alaca and K. S. Williams, Evaluation of the convolution sums ..., Journal of Number Theory, Volume 124, Issue 2, June 2007, Pages 491-510.
• E. Royer, Evaluating convolutions of divisor sums with quasimodular forms, International Journal of Number Theory 3, 2 (2007), Pages 231-261.

Crossrefs

Cf. A000385, A218277, A218278.

Programs

Maple

with(numtheory): seq((1/48)*(22*sigma[3](n) - 2*sigma[3](2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)),n=1..60); # Ridouane Oudra, Feb 23 2021

Mathematica

Table[Sum[DivisorSigma[1, k]*DivisorSigma[1, n - 2*k], {k, 1, Floor[(n - 1)/2]}], {n, 1, 50}] (* G. C. Greubel, Dec 24 2016 *)

PARI

lista(nn) = {for (i=1, nn, s = sum(m=1, floor((i-1)/2), sigma(m)*sigma(i-2*m)); print1(s , ", "););}

PARI

lista(nn) = {for (i=1, nn, v = sigma(i,3)/12 - i*sigma(i)/8 + sigma(i)/24;if (i%2 == 0, v += sigma(i/2,3)/3 - i*sigma(i/2)/4 + sigma(i/2)/24); print1(v , ", "););}

Formula

a(n) = Sum_{m < 2*n} sigma(n)*sigma(n - 2*m).

a(n) = sigma_3(n)/12 + sigma_3(n/2)/3 - n*sigma(n)/8 - n*sigma(n/2)/4 + sigma(n)/24 + sigma(n/2)/24.

a(n) = (1/48)*(22*sigma_3(n) - 2*sigma_3(2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)). - Ridouane Oudra, Feb 23 2021