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Dirichlet g.f.: (2^s - 1) / (zeta(s-1) * (2^s - 2)).
3

%I #16 Dec 02 2020 03:15:28

%S 1,-1,-3,0,-5,3,-7,0,0,5,-11,0,-13,7,15,0,-17,0,-19,0,21,11,-23,0,0,

%T 13,0,0,-29,-15,-31,0,33,17,35,0,-37,19,39,0,-41,-21,-43,0,0,23,-47,0,

%U 0,0,51,0,-53,0,55,0,57,29,-59,0,-61,31,0,0,65,-33,-67,0,69,-35,-71,0,-73,37,0

%N Dirichlet g.f.: (2^s - 1) / (zeta(s-1) * (2^s - 2)).

%C Dirichlet inverse of A000265.

%H Amiram Eldar, <a href="/A326937/b326937.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 1; a(n) = -Sum_{d|n, d<n} A000265(n/d) * a(d).

%F a(n) = A055615(n) / A006519(n).

%F Multiplicative with a(p) = -1 for p = 2 and -p for odd primes p, and a(p^e) = 0 if e > 1. - _Amiram Eldar_, Dec 02 2020

%t a[1] = 1; a[n_] := -Sum[(n/d)/2^IntegerExponent[n/d, 2] a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 75}]

%t Table[MoebiusMu[n] n/2^IntegerExponent[n, 2], {n, 1, 75}]

%t f[2, e_] := -Boole[e == 1]; f[p_, e_] := -Boole[e == 1]*p; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 02 2020 *)

%o (PARI) a(n)={n*moebius(n)/2^valuation(n,2)} \\ _Andrew Howroyd_, Oct 25 2019

%Y Cf. A000265, A006519, A013929 (positions of 0's), A030059 (positions of negative terms), A055615.

%K sign,mult

%O 1,3

%A _Ilya Gutkovskiy_, Oct 22 2019