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A319002
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Number of ordered factorizations of n where the sequence of GCDs of prime indices (A289508) of the factors is weakly increasing.
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4
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1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 5, 1, 2, 2, 8, 1, 4, 1, 5, 2, 2, 1, 12, 2, 2, 4, 5, 1, 6, 1, 16, 2, 2, 2, 11, 1, 2, 2, 12, 1, 5, 1, 5, 4, 2, 1, 28, 2, 4, 2, 5, 1, 8, 2, 12, 2, 2, 1, 18, 1, 2, 5, 32, 2, 6, 1, 5, 2, 6, 1, 29, 1, 2, 4, 5, 2, 5, 1, 28, 8, 2, 1
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OFFSET
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1,4
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COMMENTS
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Also the number of ordered multiset partitions of the multiset of prime indices of n where the sequence of GCDs of the blocks is weakly increasing. If we form a multiorder by treating integer partitions (a,...,z) as multiarrows GCD(a,...,z) <= {z,...,a}, then a(n) is the number of triangles whose composite ground is the integer partition with Heinz number n.
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LINKS
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EXAMPLE
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The a(36) = 11 ordered factorizations:
(2*2*3*3),
(2*2*9), (2*6*3), (6*2*3), (4*3*3),
(2*18), (18*2), (12*3), (4*9), (6*6),
(36).
The a(36) = 11 ordered multiset partitions:
{{1,1,2,2}}
{{1},{1,2,2}}
{{1,2,2},{1}}
{{1,1,2},{2}}
{{1,1},{2,2}}
{{1,2},{1,2}}
{{1},{1},{2,2}}
{{1},{1,2},{2}}
{{1,2},{1},{2}}
{{1,1},{2},{2}}
{{1},{1},{2},{2}}
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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]]}]];
gix[n_]:=GCD@@PrimePi/@If[n==1, {}, FactorInteger[n]][[All, 1]];
Table[Length[Select[Join@@Permutations/@facs[n], OrderedQ[gix/@#]&]], {n, 100}]
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CROSSREFS
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Cf. A000837, A007716, A001970, A056239, A063834, A289508, A296150, A316223, A317545, A318559, A319001, A319004.
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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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