A327251 - OEIS

%I #21 Aug 24 2021 17:01:29

%S 1,5,7,16,11,35,15,44,33,55,23,112,27,75,77,112,35,165,39,176,105,115,

%T 47,308,85,135,135,240,59,385,63,272,161,175,165,528,75,195,189,484,

%U 83,525,87,368,363,235,95,784,161,425,245,432,107,675,253,660,273

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

%C Inverse Moebius transform of A322577.

%C Dirichlet convolution of A001615 with A000027.

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

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

%F Multiplicative with a(p^e) = p^(e-1)*((p+1)*e + p). - _Antti Karttunen_, Aug 24 2021

%t 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

%t 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 *)

%o (PARI) mypsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615

%o a(n) = sumdiv(n, d, mypsi(n/d)*d); \\ _Michel Marcus_, Sep 15 2019

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

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Sep 15 2019