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A356945
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Number of multiset partitions of the prime indices of n such that each block covers an initial interval. Number of factorizations of n into members of A055932.
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5
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1, 1, 0, 2, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,4
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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LINKS
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EXAMPLE
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The a{n} multiset partitions for n = 8, 24, 72, 96:
{{111}} {{1112}} {{11122}} {{111112}}
{{1}{11}} {{1}{112}} {{1}{1122}} {{1}{11112}}
{{1}{1}{1}} {{11}{12}} {{11}{122}} {{11}{1112}}
{{1}{1}{12}} {{12}{112}} {{111}{112}}
{{1}{1}{122}} {{12}{1111}}
{{1}{12}{12}} {{1}{1}{1112}}
{{1}{11}{112}}
{{11}{11}{12}}
{{1}{12}{111}}
{{1}{1}{1}{112}}
{{1}{1}{11}{12}}
{{1}{1}{1}{1}{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]]}]];
nnQ[m_]:=PrimePi/@First/@FactorInteger[m]==Range[PrimePi[Max@@First/@FactorInteger[m]]];
Table[Length[Select[facs[n], And@@nnQ/@#&]], {n, 100}]
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CROSSREFS
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A000688 counts factorizations into prime powers.
A001222 counts prime factors with multiplicity.
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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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