cp's OEIS Frontend

A281119 Number of complete tree-factorizations of n >= 2.

%I A281119 #12 Nov 18 2018 19:25:11
%S A281119 1,1,1,1,1,1,2,1,1,1,3,1,1,1,5,1,3,1,3,1,1,1,9,1,1,2,3,1,4,1,12,1,1,1,
%T A281119 12,1,1,1,9,1,4,1,3,3,1,1,29,1,3,1,3,1,9,1,9,1,1,1,17,1,1,3,34,1,4,1,
%U A281119 3,1,4,1,44,1,1,3,3,1,4,1,29,5,1,1
%N A281119 Number of complete tree-factorizations of n >= 2.
%C A281119 A tree-factorization of n>=2 is either (case 1) the number n or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n into factors greater than 1. A complete (or total) tree-factorization is a tree-factorization whose leaves are all prime numbers.
%C A281119 a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Nov 18 2018
%H A281119 Michael De Vlieger, <a href="/A281119/b281119.txt">Table of n, a(n) for n = 2..10000</a>
%H A281119 A. Knopfmacher, M. Mays, <a href="http://www.mathematica-journal.com/issue/v10i1/contents/Factorizations/Factorizations_3.html">Ordered and Unordered Factorizations of Integers</a>, The Mathematica Journal 10(1), 2006.
%F A281119 a(p^n) = A196545(n) for prime p. - _Andrew Howroyd_, Nov 18 2018
%e A281119 The a(36)=12 complete tree-factorizations of 36 are:
%e A281119 (2(2(33))), (2(3(23))), (2(233)),   (3(2(23))),
%e A281119 (3(3(22))), (3(223)),   ((22)(33)), ((23)(23)),
%e A281119 (22(33)),   (23(23)),   (33(22)),   (2233).
%t A281119 postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A281119 treefacs[n_]:=If[n<=1,{{}},Prepend[Join@@Function[q,Tuples[treefacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
%t A281119 Table[Length[Select[treefacs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,2,83}]
%o A281119 (PARI) seq(n)={my(v=vector(n), w=vector(n)); v[1]=1; for(k=2, n, w[k]=v[k]+isprime(k); forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=w[k]^e*v[i]))); w[2..n]} \\ _Andrew Howroyd_, Nov 18 2018
%Y A281119 Cf. A000311, A001055, A063834, A196545, A262673, A273873, A281113, A281118.
%K A281119 nonn
%O A281119 2,7
%A A281119 _Gus Wiseman_, Jan 15 2017