A358724 - OEIS

%I #5 Dec 01 2022 08:55:36

%S 0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,1,0,2,0,2,0,0,1,0,0,1,0,

%T 1,1,0,0,1,0,0,1,0,0,2,1,1,0,1,2,1,0,0,2,2,0,1,0,0,1,1,0,2,0,2,1,0,0,

%U 2,1,0,1,1,0,3,0,1,1,0,0,3,0,1,1,2,0,1

%N Difference between the number of internal (non-leaf) nodes and the edge-height of the rooted tree with Matula-Goebel number n.

%C Edge-height (A109082) is the number of edges in the longest path from root to leaf.

%C 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.

%F a(n) = A342507(n) - A109082(n).

%e The tree (o(o)((o))(oo)) with Matula-Goebel number 210 has edge-height 3 and 5 internal nodes, so a(210) = 2.

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

%t Table[Count[MGTree[n],_[__],{0,Infinity}]-(Depth[MGTree[n]]-2),{n,100}]

%Y Positions of 0's are A209638, complement A358725.

%Y Positions of 1's are A358576, counted by A358587.

%Y Other differences: A358580, A358726, A358729.

%Y A000081 counts rooted trees, ordered A000108.

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

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

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

%Y MG core: A000040, A000720, A001222, A007097, A056239, A112798.

%Y Cf. A185650, A206487, A316321, A358577, A358578, A358592, A358730.

%K nonn

%O 1,25

%A _Gus Wiseman_, Nov 29 2022