A326826 a(n) = (1/2) * Sum_{d|n} (sigma_1(d) + sigma_2(d)), where sigma_1 = A000203 and sigma_2 = A001157.

1, 5, 8, 19, 17, 43, 30, 69, 60, 95, 68, 176, 93, 171, 166, 255, 155, 342, 192, 403, 303, 395, 278, 681, 358, 543, 490, 738, 437, 961, 498, 969, 709, 911, 720, 1476, 705, 1131, 978, 1603, 863, 1773, 948, 1732, 1440, 1643, 1130, 2634, 1284, 2110, 1648, 2391, 1433, 2882, 1706

Offset

1,2

Comments

Inverse Moebius transform applied twice to triangular numbers (A000217).

Links

Table of n, a(n) for n=1..55.

Formula

G.f.: Sum_{i>=1} Sum_{j>=1} x^(i*j) / (1 - x^(i*j))^3.

G.f.: (1/2) * Sum_{i>=1} Sum_{j>=1} j * (j + 1) * x^(i*j) / (1 - x^(i*j)).

G.f.: (1/2) * Sum_{k>=1} (sigma_1(k) + sigma_2(k)) * x^k / (1 - x^k).

Dirichlet g.f.: zeta(s)^2 * (zeta(s-1) + zeta(s-2)) / 2.

a(n) = (1/2) * Sum_{d|n} d * (d + 1) * tau(n/d), where tau = A000005.

a(n) = Sum_{d|n} A007437(d).

Sum_{k=1..n} a(k) ~ zeta(3)^2 * n^3 / 6. - Vaclav Kotesovec, Dec 11 2021

Maple

with(numtheory):
a:= n-> add(d*(d+1)*tau(n/d), d=divisors(n))/2:
seq(a(n), n=1..60);  # Alois P. Heinz, Oct 20 2019

Mathematica

Table[1/2 Sum[DivisorSigma[1, d] + DivisorSigma[2, d], {d, Divisors[n]}], {n, 1, 55}]
Table[1/2 Sum[d (d + 1) DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[Sum[x^(i j)/(1 - x^(i j))^3, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest

Prog

(PARI) a(n) = sumdiv(n, d, sigma(d)+sigma(d, 2))/2; \\ Michel Marcus, Oct 20 2019
(Magma) [(1/2)*&+[DivisorSigma(1, d)+DivisorSigma(2, d):d in Divisors(n)]:n in [1..55]]; // Marius A. Burtea, Oct 20 2019

Crossrefs

Cf. A000005, A000203, A000217, A001157, A007429, A007433, A007437, A092346, A326828.

Sequence in context: A342804 A226902 A347136 * A026595 A037233 A197539

Adjacent sequences: A326823 A326824 A326825 * A326827 A326828 A326829

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 20 2019

Status

approved