OFFSET
1,4
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.
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
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 *)
PROG
(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 , ", "); ); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 25 2012
STATUS
approved