A218277 -id:A218277 - cp's OEIS frontend

A218276 Convolution of level 2 of the divisor function.

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

Offset: 1

Michel Marcus, Oct 25 2012

nonn

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).

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.

Cf. A000385, A218277, A218278.

with(numtheory): seq((1/48)*(22*sigma[3](n) - 2*sigma[3](2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)),n=1..60); # Ridouane Oudra, Feb 23 2021

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 *)

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

lista(nn) = {for (i=1, nn, 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 , ", "););}

a(n) = Sum_{m < 2*n} sigma(n)*sigma(n - 2*m).
a(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.
a(n) = (1/48)*(22*sigma_3(n) - 2*sigma_3(2*n) + 5*sigma(n) - sigma(2*n) - 24*n*sigma(n) + 6*n*sigma(2*n)). - Ridouane Oudra, Feb 23 2021

A218278 Convolution of level 4 of the divisor function.

Original entry on oeis.org

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

Michel Marcus, Oct 25 2012

nonn

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

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.

Cf. A000385, A218276, A218277.

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

a(n) = {for (i=1, n, s = sum(m=1, floor((i-1)/4), sigma(m)*sigma(i-4*m));print1(s , ", "););}

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 , ", "););}

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

A346193 Convolution of level 5 of the divisor function.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 4, 7, 6, 15, 17, 27, 34, 36, 52, 64, 75, 91, 102, 122, 155, 169, 193, 228, 263, 276, 326, 349, 415, 430, 500, 520, 620, 681, 727, 741, 881, 880, 1090, 1020, 1192, 1178, 1375, 1513, 1590, 1557, 1809, 1756, 2274, 2024, 2323, 2245, 2626, 2865

Offset: 1

Amiram Eldar, Jul 09 2021

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Şaban Alaca and Kenneth S. Williams, Evaluation of the convolution sums ..., Journal of Number Theory, Vol. 124, No. 2 (2007) pp. 491-510.
Mathieu Lemire and Kenneth S. Williams, Evaluation of two convolution sums involving the sum of divisors function, Bulletin of the Australian Mathematical Society, Vol. 73, No. 1 (2006), pp. 107-115.
Emmanuel Royer, Evaluating convolution sums of the divisor function by quasimodular forms, International Journal of Number Theory, Vol. 3, No. 2 (2007) pp. 231-261.

Cf. A000203, A000385, A001158, A030210, A218276, A218277, A218278.

a[n_] := Sum[DivisorSigma[1, k] * DivisorSigma[1, n - 5*k], {k, 1, (n - 1)/5}]; Array[a, 100]
(* or *)
c[n_] := SeriesCoefficient[q * (QPochhammer[q] * QPochhammer[q^5])^4, {q, 0, n}]; a[n_] := 5 * DivisorSigma[3, n]/312 + If[Divisible[n, 5], 125 * DivisorSigma[3, n/5]/312, 0] - n * DivisorSigma[1, n]/20 - If[Divisible[n, 5], n * DivisorSigma[1, n/5]/4, 0] + DivisorSigma[1, n]/24 + If[Divisible[n, 5], DivisorSigma[1, n/5]/24, 0] - c[n]/130; Array[a, 100]

a(n) = Sum_{k < n/5} sigma(k) * sigma(n-5*k).

a(n) = 5*sigma_3(n)/312 + 125*sigma_3(n/5)/312 + (1/24 - n/20)*sigma(n) + (1/24 - n/4)*sigma(n/5) - c_5(n)/130, where sigma_3(n/5) = sigma(n/5) = 0 if n is not divisible by 5, and c_5(n) is the coefficient of q^n in the expansion of (eta(q) * eta(q^5))^4 (A030210).

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A218276 Convolution of level 2 of the divisor function.

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A218278 Convolution of level 4 of the divisor function.

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A346193 Convolution of level 5 of the divisor function.

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