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A305254
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Number of factorizations f of n into factors greater than 1 such that the graph of f is a forest.
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1
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1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2
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OFFSET
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1,4
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COMMENTS
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Given a factorization f consisting of positive integers greater than one, let G(F) be a multigraph with one vertex for each factor and n edges between any two vertices with n common divisors greater than 1. For example, G(6,14,15,35) is a 4-cycle; G(6,12) is a 2-cycle because 6 and 12 have multiple common divisors. This sequence counts factorizations f such that G(f) is a forest, meaning it has no cycles. [Comment edited by Robert Munafo, Mar 24 2024]
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LINKS
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EXAMPLE
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The a(72) = 14 factorizations:
(72)
(2*36) (3*24) (4*18) (8*9)
(2*2*18) (2*3*12) (2*4*9) (3*3*8) (3*4*6)
(2*2*2*9) (2*2*3*6) (2*3*3*4)
(2*2*2*3*3)
not counted: (2*6*6) because 6 and 6 share multiple divisors; likewise (6*12) because 6 and 12 share multiple divisors.
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MATHEMATICA
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zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[n], Function[f, And@@(zensity[Select[f, Function[x, Divisible[#, x]]]]==-1&/@zsm[f])]]], {n, 200}]
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CROSSREFS
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Cf. A001970, A048143, A281116, A285572, A286518, A286520, A303386, A304714, A304716, A305149, A305193, A305253.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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