A318046 a(n) is the number of initial subtrees (subtrees emanating from the root) of the unlabeled rooted tree with Matula-Goebel number n.

1, 2, 3, 2, 4, 3, 3, 2, 5, 4, 5, 3, 4, 3, 7, 2, 4, 5, 3, 4, 5, 5, 6, 3, 10, 4, 9, 3, 5, 7, 6, 2, 9, 4, 7, 5, 4, 3, 7, 4, 5, 5, 4, 5, 13, 6, 8, 3, 5, 10, 7, 4, 3, 9, 13, 3, 5, 5, 5, 7, 6, 6, 9, 2, 10, 9, 4, 4, 11, 7, 5, 5, 6, 4, 19, 3, 9, 7, 6, 4, 17, 5, 7, 5

Offset

1,2

Comments

We require that an initial subtree contain either all or none of the branchings under any given node.

Links

Table of n, a(n) for n=1..84.

Index entries for sequences related to Matula-Goebel numbers

Formula

a(1) = 1 and if n > 1 has prime factorization n = prime(x_1)^y_1 * ... * prime(x_k)^y_k then a(n) = 1 + a(x_1)^y_1 * ... * a(x_k)^y_k.

Example

70 is the Matula-Goebel number of the tree (o((o))(oo)), which has 7 distinct initial subtrees: {o, (ooo), (oo(oo)), (o(o)o), (o(o)(oo)), (o((o))o), (o((o))(oo))}. So a(70) = 7.

Mathematica

si[n_]:=If[n==1, 1, 1+Product[si[PrimePi[b[[1]]]]^b[[2]], {b, FactorInteger[n]}]];
Array[si, 100]

Crossrefs

Cf. A000081, A007097, A007853, A049076, A061773, A061775, A076146, A109082, A109129, A206491, A303431, A316476, A317713.

Sequence in context: A317713 A341041 A361660 * A246348 A205782 A070296

Adjacent sequences: A318043 A318044 A318045 * A318047 A318048 A318049

Keyword

nonn

Author

Gus Wiseman, Aug 13 2018

Status

approved