A218278 Convolution of level 4 of the divisor function.

0, 0, 0, 0, 1, 3, 4, 7, 9, 21, 20, 36, 35, 66, 52, 101, 84, 147, 120, 224, 160, 285, 220, 394, 281, 483, 360, 680, 455, 750, 560, 1025, 680, 1116, 800, 1512, 969, 1575, 1148, 2088, 1330, 2160, 1540, 2860, 1771, 2838, 2024, 3734, 2286, 3651, 2640, 4816, 2925

Offset

1,6

Comments

Named W4(n) by S. Alaca and K. S. Williams.

Links

Table of n, a(n) for n=1..53.

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) p. 231-261.

Formula

a(n) = Sum_{m<4n} sigma(n)*sigma(n-4*m).

a(n) = sigma_3(n)/48 - n*sigma(n)/16 + sigma(n)/24 + sigma_3(n/4)/3 - n*sigma(n/4)/4 + sigma(n/4)/24 + sigma_3(n/2)/16.

a(n) = (1/48)*(sigma_3(n) + 2*sigma(n) - 3*n*sigma(n)) + (1/768)*((1 + (-1)^n))*(173*sigma_3(n) - 21*sigma_3(2*n) + 28*sigma(n) - 12*sigma(2*n) - 168*n*sigma(n) + 72*n*sigma(2*n)). - Ridouane Oudra, Nov 23 2022

Maple

with(numtheory): seq(add(sigma(k)*sigma(n-4*k), k=1..floor(n/4)), n=1..70); # Ridouane Oudra, Nov 23 2022

Prog

(PARI) a(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/4), sigma(m)*sigma(i-4*m)); print1(s , ", "); ); }
(PARI) a(n) = {for (i=1, n, v = sigma(i, 3)/48 - i*sigma(i)/16 + sigma(i)/24; if (i%4 == 0, v += sigma(i/4, 3)/3 - i*sigma(i/4)/4 + sigma(i/4)/24); if (i%2 == 0, v += sigma(i/2, 3)/16); print1(v , ", "); ); }

Crossrefs

Cf. A000385, A218276, A218277.

Sequence in context: A076211 A167186 A221212 * A101062 A359394 A295399

Adjacent sequences: A218275 A218276 A218277 * A218279 A218280 A218281

Keyword

nonn

Author

Michel Marcus, Oct 25 2012

Status

approved