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A303710
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Number of factorizations of numbers that are not perfect powers using only numbers that are not perfect powers.
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6
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1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 2, 3, 1, 5, 1, 2, 2, 2, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2, 3, 2, 2, 2, 6, 1, 3, 3
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OFFSET
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1,4
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COMMENTS
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Note that a factorization of a number that is not a perfect power (A007916) is always itself aperiodic, meaning the multiplicities of its factors are relatively prime.
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LINKS
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EXAMPLE
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The a(19) = 4 factorizations of 24 are (2*2*2*3), (2*2*6), (2*12), (24).
The a(23) = 5 factorizations of 30 are (2*3*5), (2*15), (3*10), (5*6), (30).
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MATHEMATICA
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radQ[n_] := And[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[Divisors[n], radQ]}]]; Table[Length[facsr[n]], {n, Select[Range[100], radQ]}]
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CROSSREFS
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Cf. A001055, A001597, A007716, A007916, A052409, A052410, A281116, A303365, A303386, A303707, A303708, A303709.
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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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