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A218276 Convolution of level 2 of the divisor function. 3
0, 0, 1, 3, 7, 16, 22, 45, 49, 100, 95, 178, 161, 304, 250, 465, 372, 676, 525, 952, 720, 1280, 946, 1702, 1217, 2156, 1570, 2764, 1925, 3376, 2360, 4185, 2912, 4944, 3404, 6121, 4047, 6960, 4858, 8344, 5530, 9600, 6391, 11246, 7513, 12496, 8372, 14926, 9486 (list; graph; refs; listen; history; text; internal format)
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

W2(n) = Sum_{m < 2n} sigma(n)*sigma(n - 2*m).

W2(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.

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) W2(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/2), sigma(m)*sigma(i-2*m)); print1(s , ", "); ); }

(PARI) W2(n) = {for (i=1, n, 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

Cf. A000385, A218277, A218278.

Sequence in context: A162159 A190890 A116040 * A221025 A036666 A218359

Adjacent sequences:  A218273 A218274 A218275 * A218277 A218278 A218279

KEYWORD

nonn

AUTHOR

Michel Marcus, Oct 25 2012

STATUS

approved

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Last modified February 29 03:43 EST 2020. Contains 332353 sequences. (Running on oeis4.)