A295279 Number of strict tree-factorizations of n.

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 10, 1, 2, 2, 4, 1, 8, 1, 6, 2, 2, 2, 12, 1, 2, 2, 10, 1, 8, 1, 4, 4, 2, 1, 26, 1, 4, 2, 4, 1, 10, 2, 10, 2, 2, 1, 28, 1, 2, 4, 12, 2, 8, 1, 4, 2, 8, 1, 44, 1, 2, 4, 4, 2, 8, 1, 26, 3, 2, 1

Offset: 1

Author

Gus Wiseman, Nov 19 2017

Keywords

• nonn

Comments

A strict tree-factorization of n is either (case 1) the number n itself or (case 2) a set of two or more strict tree-factorizations, one of each factor in a factorization of n into distinct factors greater than one.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

Examples

			The a(30) = 8 strict tree-factorizations are: 30, (2*3*5), (2*15), (2*(3*5)), (3*10), (3*(2*5)), (5*6), (5*(2*3)).
The a(36) = 12 strict tree-factorizations are: 36, (2*3*6), (2*3*(2*3)), (2*18), (2*(2*9)), (2*(3*6)), (2*(3*(2*3))), (3*12), (3*(2*6)), (3*(2*(2*3))), (3*(3*4)), (4*9).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005804, A045778, A273873, A281113 A281118, A292504, A294786, A295281.

Programs

Mathematica

sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sft[n_]:=1+Total[Function[fac,Times@@sft/@fac]/@Select[sfs[n],Length[#]>1&]];
Array[sft,100]

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, v[j]+=w[k]*v[j/k])); w} \\ Andrew Howroyd, Nov 18 2018

Formula

a(product of n distinct primes) = A005804(n).

a(prime^n) = A273873(n).

Dirichlet g.f.: (Zeta(s) + Product_{n >= 2}(1 + a(n)/n^s))/2.