cp's OEIS Frontend

A218278 Convolution of level 4 of the divisor function.

%I A218278 #19 Nov 25 2022 13:16:48
%S A218278 0,0,0,0,1,3,4,7,9,21,20,36,35,66,52,101,84,147,120,224,160,285,220,
%T A218278 394,281,483,360,680,455,750,560,1025,680,1116,800,1512,969,1575,1148,
%U A218278 2088,1330,2160,1540,2860,1771,2838,2024,3734,2286,3651,2640,4816,2925
%N A218278 Convolution of level 4 of the divisor function.
%C A218278 Named W4(n) by S. Alaca and K. S. Williams.
%H A218278 S. Alaca and K. S. Williams, <a href="http://dx.doi.org/10.1016/j.jnt.2006.10.004">Evaluation of the convolution sums ...</a>, Journal of Number Theory, Volume 124, Issue 2, June 2007, Pages 491-510.
%H A218278 E. Royer, <a href="http://arxiv.org/abs/math/0510429">Evaluating convolutions of divisor sums with quasimodular forms</a>, arXiv:math/0510429 [math.NT], 2005-2006; International Journal of Number Theory 3, 2 (2007) p. 231-261.
%F A218278 a(n) = Sum_{m<4n} sigma(n)*sigma(n-4*m).
%F A218278 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.
%F A218278 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
%p A218278 with(numtheory): seq(add(sigma(k)*sigma(n-4*k), k=1..floor(n/4)), n=1..70); # _Ridouane Oudra_, Nov 23 2022
%o A218278 (PARI) a(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/4), sigma(m)*sigma(i-4*m));print1(s , ", "););}
%o A218278 (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 , ", "););}
%Y A218278 Cf. A000385, A218276, A218277.
%K A218278 nonn
%O A218278 1,6
%A A218278 _Michel Marcus_, Oct 25 2012