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A301829
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Number of ways to choose a nonempty submultiset of a factorization of n into factors greater than one.
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7
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0, 1, 1, 3, 1, 4, 1, 7, 3, 4, 1, 12, 1, 4, 4, 15, 1, 12, 1, 12, 4, 4, 1, 29, 3, 4, 7, 12, 1, 17, 1, 29, 4, 4, 4, 37, 1, 4, 4, 29, 1, 17, 1, 12, 12, 4, 1, 64, 3, 12, 4, 12, 1, 29, 4, 29, 4, 4, 1, 53, 1, 4, 12, 54, 4, 17, 1, 12, 4, 17, 1, 92, 1, 4, 12, 12, 4, 17
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OFFSET
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1,4
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LINKS
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FORMULA
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a(n) = Sum_{d|n, d>1} f(d) * f(n/d) where f(n) = A001055(n) is the number of factorizations of n into factors greater than 1.
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EXAMPLE
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The a(12) = 12 submultisets ("<" means subset or equal):
(2)<(2*2*3), (3)<(2*2*3), (2*2)<(2*2*3), (2*3)<(2*2*3), (2*2*3)<(2*2*3),
(2)<(2*6), (6)<(2*6), (2*6)<(2*6),
(3)<(3*4), (4)<(3*4), (3*4)<(3*4),
(12)<(12).
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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[Sum[Length[facs[d]]*Length[facs[n/d]], {d, Rest[Divisors[n]]}], {n, 100}]
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CROSSREFS
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Cf. A000712, A001055, A001222, A001405, A122768, A276024, A281113, A284640, A295632, A299701, A299702, A299729, A301830.
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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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