OFFSET
1,2
COMMENTS
A tree-factorization of n > 1 is either (case 1) the number n itself, or (case 2) a sequence of two or more tree-factorizations, one of each part of a weakly increasing factorization of n into factors > 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
The a(1) = 1 through a(4) = 23 tree-factorizations:
2 3 5 7
4 6 9
(2*2) 8 10
(2*3) 12
(2*4) 16
(2*2*2) (2*5)
(2*(2*2)) (2*6)
(2*8)
(3*3)
(3*4)
(4*4)
(2*2*3)
(2*2*4)
(2*2*2*2)
(2*(2*3))
((2*2)*4)
(2*(2*4))
(3*(2*2))
(4*(2*2))
(2*(2*2*2))
(2*2*(2*2))
((2*2)*(2*2))
(2*(2*(2*2)))
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
physemi[n_]:=Prepend[Join@@Table[Tuples[physemi/@f], {f, Select[facs[n], Length[#]>1&]}], n];
Table[Sum[Length[physemi[Times@@Prime/@m]], {m, IntegerPartitions[n]}], {n, 8}]
PROG
(PARI) \\ here TF(n) is n terms of A281118 as vector.
TF(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] += w[k]^e*v[i]))); w}
a(n)={my(v=[prod(i=1, #p, prime(p[i])) | p<-partitions(n)], tf=TF(vecmax(v))); sum(i=1, #v, tf[v[i]])} \\ Andrew Howroyd, Dec 09 2020
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jan 31 2020
EXTENSIONS
a(13)-a(20) from Andrew Howroyd, Dec 09 2020
STATUS
approved