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 A292504 Number of orderless tree-factorizations of n. 55
 1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 2, 2, 11, 1, 6, 1, 6, 2, 2, 1, 20, 2, 2, 4, 6, 1, 8, 1, 30, 2, 2, 2, 27, 1, 2, 2, 20, 1, 8, 1, 6, 6, 2, 1, 74, 2, 6, 2, 6, 1, 20, 2, 20, 2, 2, 1, 38, 1, 2, 6, 96, 2, 8, 1, 6, 2, 8, 1, 114, 1, 2, 6, 6, 2, 8, 1, 74, 11, 2, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS A factorization of n is a finite multiset of positive integers greater than 1 with product n. An orderless tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless tree-factorizations, one of each factor in a factorization of n. a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018 LINKS Michael De Vlieger, Table of n, a(n) for n = 1..16384 Michael De Vlieger, Records and indices of records. FORMULA a(p^n) = A141268(n) for prime p. - Andrew Howroyd, Nov 18 2018 EXAMPLE The a(16)=11 orderless tree-factorizations are: 16, (28), (2(24)), (2(2(22))), (2(222)), (44), (4(22)), ((22)(22)), (224), (22(22)), (2222). MATHEMATICA postfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[postfacs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]]; oltfacs[n_]:=If[n<=1, {{}}, Prepend[Union@@Function[q, Sort/@Tuples[oltfacs/@q]]/@DeleteCases[postfacs[n], {n}], n]]; Table[Length[oltfacs[n]], {n, 83}] PROG (PARI) seq(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += binomial(e+w[k]-1, e)*v[i]))); w} \\ Andrew Howroyd, Nov 18 2018 CROSSREFS Cf. A000311, A001055, A050336, A141268, A281118, A292505. Sequence in context: A050363 A308063 A166974 * A281118 A284289 A111588 Adjacent sequences:  A292501 A292502 A292503 * A292505 A292506 A292507 KEYWORD nonn AUTHOR Gus Wiseman, Sep 17 2017 STATUS approved

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Last modified July 26 02:06 EDT 2021. Contains 346294 sequences. (Running on oeis4.)