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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, 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
Sequence in context: A050363 A308063 A166974 * A281118 A349137 A284289
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 17 2017
STATUS
approved

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Last modified March 19 04:26 EDT 2024. Contains 370952 sequences. (Running on oeis4.)