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A316695 Number of series-reduced locally disjoint rooted trees whose leaves form the integer partition with Heinz number n. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..84.

EXAMPLE

The a(24) = 8 trees:

  (1(1(12)))

  (1(2(11)))

  (2(1(11)))

  (1(112))

  (2(111))

  (11(12))

  (12(11))

  (1112)

MATHEMATICA

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}]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 10 2018

STATUS

approved

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Last modified January 18 21:06 EST 2021. Contains 340262 sequences. (Running on oeis4.)