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A114592 Sum_{n>=1} a(n)/n^s = Product_{k>=2} (1 - 1/k^s).

%I A114592 #27 Jan 07 2023 10:14:02
%S A114592 1,-1,-1,-1,-1,0,-1,0,-1,0,-1,1,-1,0,0,0,-1,1,-1,1,0,0,-1,1,-1,0,0,1,
%T A114592 -1,1,-1,1,0,0,0,1,-1,0,0,1,-1,1,-1,1,1,0,-1,1,-1,1,0,1,-1,1,0,1,0,0,
%U A114592 -1,1,-1,0,1,0,0,1,-1,1,0,1,-1,1,-1,0,1,1,0,1
%N A114592 Sum_{n>=1} a(n)/n^s = Product_{k>=2} (1 - 1/k^s).
%C A114592 For n >= 2, Sum_{k|n} A001055(n/k) * a(k) = 0. A114591(n) = Sum_{k|n} a(k).
%C A114592 First entry greater than 1 in absolute value is a(360) = -2. - _Gus Wiseman_, Sep 15 2018
%H A114592 Antti Karttunen, <a href="/A114592/b114592.txt">Table of n, a(n) for n = 1..10000</a>
%F A114592 a(1) = 1; for n>= 2, a(n) = sum, over ways to factor n into any number of distinct integers >= 2, of (-1)^(number of integers in a factorization). (See example.)
%e A114592 24 can be factored into distinct integers (each >= 2) as 24; as 4*6, 3*8 and 2*12; and as 2*3*4. (A045778(24) = 5).
%e A114592 So a(24) = (-1)^1 + 3*(-1)^2 + (-1)^3 = 1, where the 1 exponent is due to the 1 factor of the 24 = 24 factorization and the 2 exponent is due to the 3 cases of 2 factors each of the 24 = 4*6 = 3*8 = 2*12 factorizations and the 3 exponent is due to the 24 = 2*3*4 factorization.
%t A114592 strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
%t A114592 Table[Sum[(-1)^Length[f],{f,strfacs[n]}],{n,100}] (* _Gus Wiseman_, Sep 15 2018 *)
%o A114592 (PARI)
%o A114592 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 A114592(n) = A114592aux(n,n); \\ _Antti Karttunen_, Jul 23 2017
%Y A114592 Cf. A001055, A045778, A114591.
%Y A114592 Cf. A001222, A162247, A190938, A259936, A281116, A303386, A316441, A319237, A319238.
%K A114592 sign
%O A114592 1,360
%A A114592 _Leroy Quet_, Dec 11 2005
%E A114592 More terms from _Antti Karttunen_, Jul 23 2017