login
A358725
Matula-Goebel numbers of rooted trees with a greater number of internal (non-leaf) vertices than edge-height.
5
9, 15, 18, 21, 23, 25, 27, 30, 33, 35, 36, 39, 42, 45, 46, 47, 49, 50, 51, 54, 55, 57, 60, 61, 63, 65, 66, 69, 70, 72, 73, 75, 77, 78, 81, 83, 84, 85, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 102, 103, 105, 108, 110, 111, 113, 114, 115, 117, 119, 120, 121
OFFSET
1,1
COMMENTS
Edge-height (A109082) is the number of edges 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.
FORMULA
A342507(a(n)) > A109082(a(n)).
EXAMPLE
The terms together with their corresponding trees begin:
9: ((o)(o))
15: ((o)((o)))
18: (o(o)(o))
21: ((o)(oo))
23: (((o)(o)))
25: (((o))((o)))
27: ((o)(o)(o))
30: (o(o)((o)))
33: ((o)(((o))))
35: (((o))(oo))
36: (oo(o)(o))
39: ((o)(o(o)))
42: (o(o)(oo))
45: ((o)(o)((o)))
46: (o((o)(o)))
47: (((o)((o))))
49: ((oo)(oo))
50: (o((o))((o)))
MATHEMATICA
MGTree[n_]:=If[n==1, {}, MGTree/@Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Count[MGTree[#], _[__], {0, Infinity}]>Depth[MGTree[#]]-2&]
CROSSREFS
Complement of A209638 (the case of equality).
These trees are counted by A316321.
Positions of positive terms in A358724.
The case of equality for node-height is A358576.
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.
Differences: A358580, A358724, A358726, A358729.
Sequence in context: A071149 A289689 A316752 * A358576 A373995 A110473
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 29 2022
STATUS
approved