A327251 Expansion of Sum_{k>=1} psi(k) * x^k / (1 - x^k)^2, where psi = A001615.

1, 5, 7, 16, 11, 35, 15, 44, 33, 55, 23, 112, 27, 75, 77, 112, 35, 165, 39, 176, 105, 115, 47, 308, 85, 135, 135, 240, 59, 385, 63, 272, 161, 175, 165, 528, 75, 195, 189, 484, 83, 525, 87, 368, 363, 235, 95, 784, 161, 425, 245, 432, 107, 675, 253, 660, 273

Offset: 1

Author

Ilya Gutkovskiy, Sep 15 2019

Keywords

• nonn
• mult

Comments

Inverse Moebius transform of A322577.

Dirichlet convolution of A001615 with A000027.

Crossrefs

Cf. A000027, A001615, A018804, A060648, A322577, A347127.

Programs

Mathematica

nmax = 57; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k] x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := p^(e - 1)*((p + 1)*e + p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)

PARI

mypsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = sumdiv(n, d, mypsi(n/d)*d); \\ Michel Marcus, Sep 15 2019

Formula

a(n) = Sum_{d|n} psi(n/d) * d.

a(p) = 2*p + 1, where p is prime.

Multiplicative with a(p^e) = p^(e-1)*((p+1)*e + p). - Antti Karttunen, Aug 24 2021