OFFSET
1,3
COMMENTS
Dirichlet inverse of A000593.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A000593(k) * A(x^k).
Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1) * (1 - 2^(1-s))).
a(1) = 1; a(n) = -Sum_{d|n, d<n} A000593(n/d) * a(d).
Multiplicative with a(2^e) = -1 if e = 1 and 0 otherwise, and a(p^e) = -(p+1) if e = 1, p if e = 2 and 0 otherwise, for an odd prime p. - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} abs(a(k)) ~ 30*n^2/Pi^4. - Vaclav Kotesovec, May 30 2024
MATHEMATICA
Table[DivisorSum[n, # MoebiusMu[#] MoebiusMu[n/#] &, OddQ[#] &], {n, 1, 69}]
a[n_] := If[n == 1, n, -Sum[If[d < n, DivisorSum[n/d, Mod[#, 2] # &] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 69}]
f[p_, e_] := If[p == 2, -Boole[e == 1], Which[e == 1, -p - 1, e == 2, p, e > 2, 0]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)
PROG
(Magma) [&+[d*MoebiusMu(d)*MoebiusMu(n div d): d in [a:a in Divisors(n)| IsOdd(a)]]:n in [1..70]]; // Marius A. Burtea, Sep 15 2019
CROSSREFS
KEYWORD
sign,mult,easy
AUTHOR
Ilya Gutkovskiy, Sep 15 2019
STATUS
approved