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

1,6

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

Links

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

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.

Example

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).

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]

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

Crossrefs

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

Sequence in context: A295636 A050334 A342084 * A316784 A284974 A293222

Adjacent sequences: A295276 A295277 A295278 * A295280 A295281 A295282

Keyword

nonn

Author

Gus Wiseman, Nov 19 2017

Status

approved