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A composite analog of the Moebius function: Sum_{n>=1} a(n)/n^s = Product_{c=composites} (1 - 1/c^s) = zeta(s) *Product_{k>=2} (1 - 1/k^s).
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%I #22 Jul 23 2017 21:46:36

%S 1,0,0,-1,0,-1,0,-1,-1,-1,0,-1,0,-1,-1,-1,0,-1,0,-1,-1,-1,0,0,-1,-1,

%T -1,-1,0,-1,0,0,-1,-1,-1,0,0,-1,-1,0,0,-1,0,-1,-1,-1,0,1,-1,-1,-1,-1,

%U 0,0,-1,0,-1,-1,0,1,0,-1,-1,0,-1,-1,0,-1,-1,-1,0,2,0,-1,-1,-1,-1,-1,0,1

%N A composite analog of the Moebius function: Sum_{n>=1} a(n)/n^s = Product_{c=composites} (1 - 1/c^s) = zeta(s) *Product_{k>=2} (1 - 1/k^s).

%C For n >= 2, Sum_{k|n} A050370(n/k) * a(k) = 0.

%C Sum_{n>=1} a(n)/n^2 = Pi^2/12.

%C a(n) = Sum_{k|n} A114592(k).

%H Antti Karttunen, <a href="/A114591/b114591.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(1) = 1; for n>= 2, a(n) = sum, over ways to factor n into any number of distinct composites, of (-1)^(number of composites in a factorization). (See example.)

%e 24 can be factored into distinct composites as 24 and as 4*6.

%e So a(24) = (-1)^1 + (-1)^2 = 0, where the 1 exponent is due to the 1 factor of the 24 = 24 factorization and the 2 exponent is due to the 2 factors of the 24 = 4*6 factorization.

%t a[n_] := Total[((-1)^Length[#]& ) /@ Select[Subsets[Select[Rest[Divisors[n]], !PrimeQ[#]& ]], Times @@ # == n & ]]; Table[a[n], {n, 1, 80}]

%o (PARI)

%o A114592aux(n, k) = if(1==n, 1, sumdiv(n, d, if(d > 1 && d <= k && d < n, (-1)*A114592aux(n/d, d-1))) - (n<=k)); \\ After code in A045778.

%o A114592(n) = A114592aux(n,n);

%o A114591(n) = sumdiv(n,d,A114592(d)); \\ _Antti Karttunen_, Jul 23 2017

%Y Cf. A045778, A050370, A114592.

%K sign

%O 1,72

%A _Leroy Quet_, Dec 11 2005

%E More terms from _Jean-François Alcover_, Sep 26 2013