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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

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-Fran├žois 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

Cf. A000385, A218276, A218278.

Sequence in context: A048155 A242342 A204292 * A103080 A055720 A054184

Adjacent sequences:  A218274 A218275 A218276 * A218278 A218279 A218280

KEYWORD

nonn

AUTHOR

Michel Marcus, Oct 25 2012

STATUS

approved

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Last modified August 17 23:13 EDT 2022. Contains 356204 sequences. (Running on oeis4.)