A341635 - OEIS

%I #17 Feb 19 2021 08:33:42

%S 1,-2,-3,1,-5,6,-7,0,2,10,-11,-3,-13,14,15,0,-17,-4,-19,-5,21,22,-23,

%T 0,4,26,0,-7,-29,-30,-31,0,33,34,35,2,-37,38,39,0,-41,-42,-43,-11,-10,

%U 46,-47,0,6,-8,51,-13,-53,0,55,0,57,58,-59,15,-61,62,-14,0,65

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

%C Dirichlet inverse of A003967.

%C Moebius transform of A097945.

%C From _Vaclav Kotesovec_, Feb 19 2021: (Start)

%C Abs(a(n)) <= n.

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

%H Amiram Eldar, <a href="/A341635/b341635.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F Multiplicative with a(p^e) = -p if e=1, p-1 if e=2, and 0 otherwise. - _Amiram Eldar_, Feb 19 2021

%t Table[Sum[EulerPhi[d] MoebiusMu[d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 65}]

%t Table[Sum[MoebiusMu[GCD[n, k]] MoebiusMu[n/GCD[n, k]], {k, n}], {n, 65}]

%o (PARI) a(n) = sumdiv(n, d, eulerphi(d)*moebius(d)*moebius(n/d)); \\ _Michel Marcus_, Feb 17 2021

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

%K sign,mult

%O 1,2

%A _Ilya Gutkovskiy_, Feb 16 2021