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A319118
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Number of multimin tree-factorizations of n.
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3
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1, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 8, 1, 2, 2, 24, 1, 6, 1, 8, 2, 2, 1, 42, 2, 2, 6, 8, 1, 8, 1, 112, 2, 2, 2, 38, 1, 2, 2, 42, 1, 8, 1, 8, 8, 2, 1, 244, 2, 6, 2, 8, 1, 24, 2, 42, 2, 2, 1, 58, 1, 2, 8, 568, 2, 8, 1, 8, 2, 8, 1, 268, 1, 2, 6, 8, 2, 8, 1, 244, 24
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OFFSET
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1,4
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COMMENTS
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A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of multimin tree-factorizations, one of each factor in a multimin factorization of n with at least two factors.
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LINKS
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FORMULA
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a(product of n distinct primes) = A005804(n).
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EXAMPLE
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The a(12) = 8 multimin tree-factorizations:
12,
(2*6), (4*3), (6*2), (2*2*3),
(2*(2*3)), ((2*2)*3), ((2*3)*2).
Or as series-reduced plane trees of multisets:
112,
(1,12), (11,2), (12,1), (1,1,2),
(1,(1,2)), ((1,1),2)), ((1,2),1).
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n], Length[#]>1&], OrderedQ[FactorInteger[#][[1, 1]]&/@#]&]), n];
Table[Length[mmftrees[n]], {n, 100}]
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CROSSREFS
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Cf. A001055, A005804, A020639, A118376, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A318577, A319119.
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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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