A326828 a(n) = (1/2) * Sum_{d|n} mu(n/d) * phi(d) * (psi(d) + 1), where mu = A008683, phi = A000010 and psi = A001615.

1, 1, 4, 5, 13, 7, 26, 19, 34, 23, 64, 32, 89, 47, 82, 74, 151, 64, 188, 105, 167, 119, 274, 127, 296, 167, 294, 214, 433, 161, 494, 292, 421, 287, 548, 290, 701, 359, 590, 417, 859, 329, 944, 540, 742, 527, 1126, 506, 1170, 576, 1012, 757, 1429, 576, 1382

Offset

1,3

Comments

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} 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

Cf. A000010, A000217, A001615, A007427, A007431, A007438, A008683, A321322, A326826.

Sequence in context: A274077 A028272 A003969 * A348398 A132140 A102703

Adjacent sequences: A326825 A326826 A326827 * A326829 A326830 A326831

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 20 2019

Status

approved