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A370645
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Number of integer factorizations of n into unordered factors > 1 such that only one set can be obtained by choosing a different prime factor of each factor.
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9
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1
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OFFSET
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1,12
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COMMENTS
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All of these factorizations are co-balanced (A340596).
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LINKS
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EXAMPLE
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The factorization f = (3*6*10) has prime factor choices (3,2,2), (3,3,2), (3,2,5), and (3,3,5), of which only (3,2,5) has all different parts, so f is counted under a(180).
The a(n) factorizations for n = 2, 12, 24, 36, 72, 120, 144, 180, 288:
(2) (2*6) (3*8) (4*9) (8*9) (3*5*8) (2*72) (4*5*9) (3*96)
(3*4) (4*6) (6*6) (2*36) (4*5*6) (3*48) (5*6*6) (4*72)
(2*12) (2*18) (3*24) (2*3*20) (4*36) (2*3*30) (6*48)
(3*12) (4*18) (2*5*12) (6*24) (2*5*18) (8*36)
(6*12) (2*6*10) (8*18) (2*6*15) (9*32)
(3*4*10) (9*16) (2*9*10) (12*24)
(12*12) (3*4*15) (16*18)
(3*5*12) (2*144)
(3*6*10)
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join @@ Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Length[Union[Sort/@Select[Tuples[First /@ FactorInteger[#]&/@#], UnsameQ@@#&]]]==1&]], {n, 100}]
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CROSSREFS
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Maximal sets of this type are counted by A370585.
A355741 counts ways to choose a prime factor of each prime index.
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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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