A332712 - OEIS

%I #14 Nov 30 2020 03:57:57

%S 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,

%T 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,

%U 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

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

%H Amiram Eldar, <a href="/A332712/b332712.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

%F 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

%F Multiplicative with a(p^e) = A028242(e). - _Amiram Eldar_, Nov 30 2020

%t Table[Sum[MoebiusMu[d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 100}]

%t A005361[n_] := Times @@ (#[[2]] & /@ FactorInteger[n]); a[n_] := Sum[(-1)^PrimeOmega[n/d] A005361[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]

%t 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 *)

%o (PARI) a(n) = sumdiv(n, d, moebius(d/gcd(d, n/d))); \\ _Michel Marcus_, Feb 20 2020

%Y 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.

%K nonn,mult

%O 1,4

%A _Ilya Gutkovskiy_, Feb 20 2020