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A303707
Number of factorizations of n using elements of A007916 (numbers that are not perfect powers).
31
1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2
OFFSET
1,6
COMMENTS
First differs from A081707 at a(60) = 9, A081707(60) = 8.
FORMULA
Dirichlet g.f.: Product_{n in A007916} 1/(1 - n^s).
EXAMPLE
The a(60) = 9 factorizations are (2*2*3*5), (2*2*15), (2*3*10), (2*5*6), (2*30), (3*20), (5*12), (6*10), (60).
MATHEMATICA
radQ[n_]:=Or[n===1, GCD@@FactorInteger[n][[All, 2]]===1];
facsr[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsr[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], radQ]}]];
Table[Length[facsr[n]], {n, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 29 2018
STATUS
approved