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A332712
a(n) = Sum_{d|n} mu(d/gcd(d, n/d)).
2
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
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
KEYWORD
nonn,mult
AUTHOR
Ilya Gutkovskiy, Feb 20 2020
STATUS
approved