OFFSET
1,3
COMMENTS
Moebius transform applied twice to triangular numbers (A000217).
FORMULA
G.f.: Sum_{i>=1} Sum_{j>=1} mu(j) * mu(i) * x^(i*j) / (1 - x^(i*j))^3.
Dirichlet g.f.: (zeta(s-1) + zeta(s-2)) / (2 * zeta(s)^2).
a(n) = (1/2) * Sum_{d|n} mu(n/d) * (phi(d) + J_2(d)), where J_2 = A007434.
a(n) = (1/2) * Sum_{d|n} d * (d + 1) * A007427(n/d).
a(n) = Sum_{d|n} mu(n/d) * A007438(d).
Sum_{k=1..n} a(k) ~ n^3 / (6*zeta(3)^2). - Vaclav Kotesovec, Dec 11 2021
MAPLE
with(numtheory):
b:= proc(n) option remember;
add(mobius(n/d)*d*(d+1)/2, d=divisors(n))
end:
a:= proc(n) option remember;
add(mobius(n/d)*b(d), d=divisors(n))
end:
seq(a(n), n=1..60); # Alois P. Heinz, Oct 20 2019
MATHEMATICA
Table[1/2 Sum[MoebiusMu[n/d] EulerPhi[d] (DirichletConvolve[j, MoebiusMu[j]^2, j, d] + 1), {d, Divisors[n]}], {n, 1, 55}]
Table[1/2 Sum[d (d + 1) DivisorSum[n/d, MoebiusMu[#] MoebiusMu[(n/d)/#] &], {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[Sum[MoebiusMu[j] MoebiusMu[i] x^(i j)/(1 - x^(i j))^3, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 20 2019
STATUS
approved