A218277 Convolution of level 3 of the divisor function.

0, 0, 0, 1, 3, 4, 10, 15, 24, 33, 45, 65, 77, 102, 143, 155, 180, 268, 255, 315, 434, 435, 462, 695, 593, 735, 960, 918, 945, 1437, 1160, 1395, 1825, 1692, 1668, 2549, 1995, 2385, 3073, 2775, 2730, 4190, 3157, 3747, 4739, 4290, 4140, 6355, 4686, 5523, 7044

Offset: 1

Author

Michel Marcus, Oct 25 2012

Keywords

• nonn

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.

Crossrefs

Cf. A000385, A218276, A218278.

Programs

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 *)

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 , ", "););}

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