|
|
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
|
|
|
FORMULA
|
|
|
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
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|