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A218277 Convolution of level 3 of the divisor function. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Named W3(n) by S. Alaca and K. S. Williams.
LINKS
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
Sequence in context: A048155 A242342 A204292 * A103080 A055720 A054184
KEYWORD
nonn
AUTHOR
Michel Marcus, Oct 25 2012
STATUS
approved

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Last modified March 28 11:44 EDT 2024. Contains 371241 sequences. (Running on oeis4.)