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 A316655 Number of series-reduced rooted trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n. 17
 0, 1, 1, 1, 2, 3, 5, 4, 12, 9, 12, 17, 33, 29, 44, 26, 90, 90, 261, 68, 168, 93, 766, 144, 197, 307, 575, 269, 2312, 428, 7068, 236, 625, 1017, 863, 954, 21965, 3409, 2342, 712 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS A rooted tree is series-reduced if every non-leaf node has at least two branches. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). LINKS FORMULA a(prime(n)) = A000669(n). a(2^n) = A000311(n). EXAMPLE Sequence of sets of trees begins: 1: 2: 1 3: (11) 4: (12) 5: (1(11)), (111) 6: (1(12)), (2(11)), (112) 7: (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111) 8: (1(23)), (2(13)), (3(12)), (123) 9: (1(1(22))), (1(2(12))), (1(122)), (2(1(12))), (2(2(11))), (2(112)), ((11)(22)), ((12)(12)), (11(22)), (12(12)), (22(11)), (1122) 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, Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])]]; Table[Length[gro[Flatten[MapIndexed[Table[#2, {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]]]], {n, 20}] CROSSREFS Cf. A000081, A000311, A000669, A001678, A005804, A056239, A141268, A181821, A292504, A296150, A300660, A304660. Cf. A316651, A316652, A316653, A316654, A316656. Sequence in context: A258861 A171038 A023395 * A318848 A193798 A101409 Adjacent sequences:  A316652 A316653 A316654 * A316656 A316657 A316658 KEYWORD nonn,more AUTHOR Gus Wiseman, Jul 09 2018 EXTENSIONS a(37)-a(40) from Robert Price, Sep 13 2018 STATUS approved

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Last modified December 10 10:58 EST 2018. Contains 318047 sequences. (Running on oeis4.)