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 A316656 Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n. 10
 0, 1, 0, 1, 0, 1, 0, 4, 3, 1, 0, 9, 0, 1, 6, 26, 0, 36, 0, 16, 10, 1, 0, 92, 21, 1, 197, 25, 0, 100, 0, 236, 15, 1, 53, 474 (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 an identity tree if no branch appears multiple times under the same root. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). LINKS FORMULA a(prime(n>1)) = 0. a(2^n) = A000311(n). EXAMPLE Sequence of sets of trees begins:    1:    2: 1    3:    4: (12)    5:    6: (1(12))    7:    8: (1(23)), (2(13)), (3(12)), (123)    9: (1(2(12))), (2(1(12))), (12(12))   10: (1(1(12)))   11:   12: (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), ((12)(13)), (12(13)), (13(12)) 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]]]]; gro[m_]:=If[Length[m]==1, m, Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])], UnsameQ@@#&]]; Table[Length[gro[Flatten[MapIndexed[Table[#2, {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]]]], {n, 30}] CROSSREFS Cf. A000081, A000311, A000669, A001678, A004111, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660. Cf. A316651, A316652, A316653. A316654, A316655. Sequence in context: A021236 A136590 A117026 * A083904 A215861 A327366 Adjacent sequences:  A316653 A316654 A316655 * A316657 A316658 A316659 KEYWORD nonn,more AUTHOR Gus Wiseman, Jul 09 2018 STATUS approved

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Last modified April 8 14:29 EDT 2020. Contains 333314 sequences. (Running on oeis4.)