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A316695
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Number of series-reduced locally disjoint rooted trees whose leaves form the integer partition with Heinz number n.
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1
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0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 8, 1, 1, 2, 3, 1, 4, 1, 10, 1, 1, 1, 12, 1, 1, 1, 8, 1, 4, 1, 3, 3, 1, 1, 23, 1, 3, 1, 3, 1, 8, 1, 8, 1, 1, 1, 16, 1, 1, 3, 24, 1, 4, 1, 3, 1, 4, 1, 37, 1, 1, 3, 3, 1, 4, 1, 23, 5, 1, 1, 16
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OFFSET
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1,8
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COMMENTS
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A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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LINKS
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Table of n, a(n) for n=1..84.
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EXAMPLE
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The a(24) = 8 trees:
(1(1(12)))
(1(2(11)))
(2(1(11)))
(1(112))
(2(111))
(11(12))
(12(11))
(1112)
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MATHEMATICA
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sps[{}]:={{}};
sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
gro[m_]:=gro[m]=If[Length[m]==1, List/@m, Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])]];
Table[Length[Select[gro[If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]], And@@Cases[#, q:{__List}:>disjointQ[q], {0, Infinity}]&]], {n, 100}]
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CROSSREFS
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Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316471, A316651, A316652, A316655.
Sequence in context: A326840 A326153 A199515 * A316767 A292505 A281119
Adjacent sequences: A316692 A316693 A316694 * A316696 A316697 A316698
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Jul 10 2018
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STATUS
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approved
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