%I #17 Sep 08 2022 08:46:24

%S 1,5,8,19,17,43,30,69,60,95,68,176,93,171,166,255,155,342,192,403,303,

%T 395,278,681,358,543,490,738,437,961,498,969,709,911,720,1476,705,

%U 1131,978,1603,863,1773,948,1732,1440,1643,1130,2634,1284,2110,1648,2391,1433,2882,1706

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

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

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

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

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

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

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

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

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

%p with(numtheory):

%p a:= n-> add(d*(d+1)*tau(n/d), d=divisors(n))/2:

%p seq(a(n), n=1..60); # _Alois P. Heinz_, Oct 20 2019

%t Table[1/2 Sum[DivisorSigma[1, d] + DivisorSigma[2, d], {d, Divisors[n]}], {n, 1, 55}]

%t Table[1/2 Sum[d (d + 1) DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 1, 55}]

%t 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

%o (PARI) a(n) = sumdiv(n, d, sigma(d)+sigma(d, 2))/2; \\ _Michel Marcus_, Oct 20 2019

%o (Magma) [(1/2)*&+[DivisorSigma(1,d)+DivisorSigma(2,d):d in Divisors(n)]:n in [1..55]]; // _Marius A. Burtea_, Oct 20 2019

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

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Oct 20 2019