A341635 a(n) = Sum_{d|n} phi(d) * mu(d) * mu(n/d).

1, -2, -3, 1, -5, 6, -7, 0, 2, 10, -11, -3, -13, 14, 15, 0, -17, -4, -19, -5, 21, 22, -23, 0, 4, 26, 0, -7, -29, -30, -31, 0, 33, 34, 35, 2, -37, 38, 39, 0, -41, -42, -43, -11, -10, 46, -47, 0, 6, -8, 51, -13, -53, 0, 55, 0, 57, 58, -59, 15, -61, 62, -14, 0, 65

Offset: 1

Author

Ilya Gutkovskiy, Feb 16 2021

Keywords

• sign
• mult

Comments

Dirichlet inverse of A003967.

Moebius transform of A097945.

From Vaclav Kotesovec, Feb 19 2021: (Start)

Abs(a(n)) <= n.

a(n) = n iff n is in A030229. (End)

Links

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

Crossrefs

Cf. A000010, A003967, A007427, A007431, A008683, A030229 (fixed points), A046099 (positions of 0's), A068341, A097945, A276833.

Programs

Mathematica

Table[Sum[EulerPhi[d] MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 65}]
Table[Sum[MoebiusMu[GCD[n, k]] MoebiusMu[n/GCD[n, k]], {k, n}], {n, 65}]

PARI

a(n) = sumdiv(n, d, eulerphi(d)*moebius(d)*moebius(n/d)); \\ Michel Marcus, Feb 17 2021

Formula

a(n) = Sum_{k=1..n} mu(gcd(n,k)) * mu(n/gcd(n,k)).

a(1) = 1; a(n) = -Sum_{d|n, d < n} A003967(n/d) * a(d).

a(n) = Sum_{d|n} mu(n/d) * A097945(d).

Multiplicative with a(p^e) = -p if e=1, p-1 if e=2, and 0 otherwise. - Amiram Eldar, Feb 19 2021