A358726 - OEIS

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A358726

Difference between the node-height and the number of leaves in the rooted tree with Matula-Goebel number n.

0, 1, 2, 0, 3, 1, 1, -1, 1, 2, 4, 0, 2, 0, 2, -2, 2, 0, 0, 1, 0, 3, 2, -1, 2, 1, 0, -1, 3, 1, 5, -3, 3, 1, 1, -1, 1, -1, 1, 0, 3, -1, 1, 2, 1, 1, 3, -2, -1, 1, 1, 0, -1, -1, 3, -2, -1, 2, 3, 0, 1, 4, -1, -4, 1, 2, 1, 0, 1, 0, 2, -2, 1, 0, 1, -2, 2, 0, 4, -1

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OFFSET

1,3

COMMENTS

Node-height is the number of nodes in the longest path from root to leaf.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

LINKS

Table of n, a(n) for n=1..80.

FORMULA

a(n) = A358552(n) - A109129(n).

EXAMPLE

The tree (oo(oo(o))) with Matula-Goebel number 148 has node-height 4 and 5 leaves, so a(148) = -1.

MATHEMATICA

MGTree[n_]:=If[n==1, {}, MGTree/@Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[(Depth[MGTree[n]]-1)-Count[MGTree[n], {}, {0, Infinity}], {n, 1000}]

CROSSREFS

Positions of first appearances are A007097 and latter terms of A000079.

Positions of 0's are A358577.