OFFSET
1,5
COMMENTS
Named W3(n) by S. Alaca and K. S. Williams.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
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, arXiv:math/0510429 [math.NT], 2005-2006; International Journal of Number Theory 3, 2 (2007), Pages 231-261.
FORMULA
a(n) = Sum_{m<3n} sigma(m)*sigma(n-3*m).
a(n) = sigma3(n)/24 - n*sigma(n)/12 + sigma(n)/24 + 3*sigma3(n/3)/8 - n*sigma(n/3)/4 + sigma(n/3)/24.
a(n) = (1/72)*(31*sigma_3(n) - sigma_3(3*n) + 7*sigma(n) - sigma(3*n) - 30*n*sigma(n) + 6*n*sigma(3*n)). - Ridouane Oudra, Mar 21 2021
MAPLE
f:= n -> add(numtheory:-sigma(m)*numtheory:-sigma(n-3*m), m=1..floor((n-1)/3)):
map(f, [$1..50]); # Robert Israel, Jun 28 2018
with(numtheory): seq((1/72)*(31*sigma[3](n) - sigma[3](3*n) + 7*sigma(n) - sigma(3*n) - 30*n*sigma(n) + 6*n*sigma(3*n)), n=1..50); # Ridouane Oudra, Mar 21 2021
MATHEMATICA
a[n_] := Sum[DivisorSigma[1, m] DivisorSigma[1, n-3m], {m, 1, (n-1)/3}];
Array[a, 50] (* Jean-François Alcover, Sep 19 2018 *)
PROG
(PARI) lista(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/3), sigma(m)*sigma(i-3*m)); print1(s , ", "); ); }
(PARI) lista(n) = {for (i=1, n, v = sigma(i, 3)/24 - i*sigma(i)/12 + sigma(i)/24; if (i%3 == 0, v += 3*sigma(i/3, 3)/8 - i*sigma(i/3)/4 + sigma(i/3)/24); print1(v , ", "); ); }
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 25 2012
STATUS
approved