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A050372
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Number of ways to factor n into distinct composite factors.
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3
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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, 2, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 1, 1, 1, 1, 0, 3, 1, 1, 0, 3, 1, 1, 1, 2, 0, 3, 1, 1, 1, 1, 1, 4, 0, 1, 1, 2, 0, 1
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OFFSET
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1,24
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COMMENTS
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a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
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LINKS
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FORMULA
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Dirichlet g.f.: Product_{n is composite}(1+1/n^s).
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MAPLE
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with(numtheory):
b:= proc(n, k) option remember;
`if`(isprime(n), 0, `if`(n>k, 0, 1)+
add(`if`(d>k or isprime(d), 0, b(n/d, d-1))
, d=divisors(n) minus {1, n}))
end:
a:= n-> b(n$2):
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MATHEMATICA
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b[n_, k_] := b[n, k] = If[PrimeQ[n], 0, If[n>k, 0, 1] + Sum[If[d>k || PrimeQ[d], 0, b[n/d, d-1]], {d, Divisors[n] ~Complement~ {1, n}}]];
a[n_] := b[n, n];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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