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A114591
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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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4
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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, -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, 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
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OFFSET
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1,72
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COMMENTS
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For n >= 2, Sum_{k|n} A050370(n/k) * a(k) = 0.
Sum_{n>=1} a(n)/n^2 = Pi^2/12.
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LINKS
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FORMULA
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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.)
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EXAMPLE
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24 can be factored into distinct composites as 24 and as 4*6.
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.
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MATHEMATICA
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a[n_] := Total[((-1)^Length[#]& ) /@ Select[Subsets[Select[Rest[Divisors[n]], !PrimeQ[#]& ]], Times @@ # == n & ]]; Table[a[n], {n, 1, 80}]
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PROG
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(PARI)
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.
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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