A309324 Expansion of Sum_{k>=1} psi(k) * x^k/(1 + x^k), where psi = Dedekind psi function (A001615).

1, 2, 5, 2, 7, 10, 9, 2, 17, 14, 13, 10, 15, 18, 35, 2, 19, 34, 21, 14, 45, 26, 25, 10, 37, 30, 53, 18, 31, 70, 33, 2, 65, 38, 63, 34, 39, 42, 75, 14, 43, 90, 45, 26, 119, 50, 49, 10, 65, 74, 95, 30, 55, 106, 91, 18, 105, 62, 61, 70, 63, 66, 153, 2, 105, 130, 69, 38, 125, 126, 73

Offset: 1

Author

Ilya Gutkovskiy, Jul 23 2019

Keywords

• nonn
• mult

Comments

Dirichlet convolution of sum of odd divisors function with characteristic function of squarefree numbers.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000593, A001615, A008966, A060648, A193356.

Programs

Mathematica

nmax = 71; CoefficientList[Series[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, k]  x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[MoebiusMu[n/d]^2 Plus @@ Select[Divisors@ d, OddQ], {d, Divisors[n]}], {n, 1, 71}]
f[2, e_] := 2; f[p_, e_] := (p^e*(p+1)-2)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 01 2020 *)

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 2, (f[i,1]^f[i,2]*(f[i,1]+1)-2)/(f[i,1]-1)));} \\ Amiram Eldar, Nov 06 2022

Formula

a(n) = Sum_{d|n} (-1)^(n/d+1) * psi(d).

a(n) = Sum_{d|n} mu(n/d)^2 * A000593(d).

Multiplicative with a(2^e) = 2, and a(p^e) = (p^e*(p+1)-2)/(p-1) for odd primes p. - Amiram Eldar, Dec 01 2020

Sum_{k=1..n} a(k) ~ (5/8) * n^2. - Amiram Eldar, Nov 06 2022