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

Offset: 1

Author

Gus Wiseman, Nov 29 2022

Keywords

• sign

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.

Examples

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

Crossrefs

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

Positions of 0's are A358577.

Other differences: A358580, A358724, A358729.

A000081 counts rooted trees, ordered A000108.

A034781 counts rooted trees by nodes and height, ordered A080936.

A055277 counts rooted trees by nodes and leaves, ordered A001263.

MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.

MG core: A000040, A000720, A001222, A056239, A112798.

Cf. A185650, A206487, A209638, A358576, A358578, A358587, A358730.

Programs

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}]

Formula

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