OFFSET
1,2
COMMENTS
Inverse Moebius transform applied twice to triangular numbers (A000217).
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 20 2019
STATUS
approved