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A281119
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Number of complete tree-factorizations of n >= 2.
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15
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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, 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, 3, 1, 4, 1, 44, 1, 1, 3, 3, 1, 4, 1, 29, 5, 1, 1
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OFFSET
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2,7
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COMMENTS
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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.
a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018
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LINKS
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FORMULA
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EXAMPLE
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The a(36)=12 complete tree-factorizations of 36 are:
(2(2(33))), (2(3(23))), (2(233)), (3(2(23))),
(3(3(22))), (3(223)), ((22)(33)), ((23)(23)),
(22(33)), (23(23)), (33(22)), (2233).
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MATHEMATICA
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postfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[postfacs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
treefacs[n_]:=If[n<=1, {{}}, Prepend[Join@@Function[q, Tuples[treefacs/@q]]/@DeleteCases[postfacs[n], {n}], n]];
Table[Length[Select[treefacs[n], FreeQ[#, _Integer?(!PrimeQ[#]&)]&]], {n, 2, 83}]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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