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

1,2

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A246341 A246355 A016580 * A220072 A065223 A248154

Adjacent sequences: A309321 A309322 A309323 * A309325 A309326 A309327

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Jul 23 2019

Status

approved