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
Keywords
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) p. 231-261.
Programs
-
Maple
with(numtheory): seq(add(sigma(k)*sigma(n-4*k), k=1..floor(n/4)), n=1..70); # Ridouane Oudra, Nov 23 2022
-
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 , ", "););}
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
Comments