login
A327278
a(n) = Sum_{d|n, d odd} d * mu(d) * mu(n/d).
7
1, -1, -4, 0, -6, 4, -8, 0, 3, 6, -12, 0, -14, 8, 24, 0, -18, -3, -20, 0, 32, 12, -24, 0, 5, 14, 0, 0, -30, -24, -32, 0, 48, 18, 48, 0, -38, 20, 56, 0, -42, -32, -44, 0, -18, 24, -48, 0, 7, -5, 72, 0, -54, 0, 72, 0, 80, 30, -60, 0, -62, 32, -24, 0, 84, -48, -68, 0, 96
OFFSET
1,3
COMMENTS
Dirichlet inverse of A000593.
LINKS
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).
a(n) = Sum_{d|n} A067856(n/d) * A055615(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
KEYWORD
sign,mult,easy
AUTHOR
Ilya Gutkovskiy, Sep 15 2019
STATUS
approved