A332712 a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).

1, 0, 0, 2, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4

Offset

1,4

Links

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

Formula

Dirichlet g.f.: zeta(2*s)^2 * zeta(3*s) / zeta(6*s).

a(n) = Sum_{d|n} mu(lcm(d, n/d)/d).

a(n) = Sum_{d|n} (-1)^bigomega(n/d) * A005361(d).

a(n) = Sum_{d|n} A010052(n/d) * A112526(d).

Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)*log(n)/(2*zeta(3)) + ((2*gamma - 1)*zeta(3/2) + 3*zeta'(3/2)/2 - 3*zeta(3/2)*zeta'(3)/zeta(3)) * sqrt(n)/zeta(3) + 6*zeta(2/3)^2 * n^(1/3)/Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 21 2020

Multiplicative with a(p^e) = A028242(e). - Amiram Eldar, Nov 30 2020

Mathematica

Table[Sum[MoebiusMu[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 100}]
A005361[n_] := Times @@ (#[[2]] & /@ FactorInteger[n]); a[n_] := Sum[(-1)^PrimeOmega[n/d] A005361[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
f[p_, e_] := 3*Floor[e/2] - e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

Prog

(PARI) a(n) = sumdiv(n, d, moebius(d/gcd(d, n/d))); \\ Michel Marcus, Feb 20 2020

Crossrefs

Cf. A001222, A001694 (positions of nonzero terms), A005361, A007427, A008683, A008836, A028242, A052485 (positions of 0's), A062838 (positions of 1's), A112526, A252505, A322483, A332685, A332713.

Sequence in context: A035699 A132406 A197881 * A079126 A339086 A186336

Adjacent sequences: A332709 A332710 A332711 * A332713 A332714 A332715

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Feb 20 2020

Status

approved