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A358576
Matula-Goebel numbers of rooted trees whose node-height equals their number of internal (non-leaf) nodes.
20
9, 15, 18, 21, 23, 30, 33, 35, 36, 39, 42, 46, 47, 49, 51, 57, 60, 61, 66, 70, 72, 73, 77, 78, 83, 84, 87, 91, 92, 93, 94, 95, 98, 102, 111, 113, 114, 119, 120, 122, 123, 129, 132, 133, 137, 140, 144, 146, 149, 151, 154, 156, 159, 166, 167, 168, 174, 177, 181
OFFSET
1,1
COMMENTS
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.
Node-height is the number of nodes in the longest path from root to leaf.
FORMULA
A358552(a(n)) = A342507(a(n)).
EXAMPLE
The terms together with their corresponding rooted trees begin:
9: ((o)(o))
15: ((o)((o)))
18: (o(o)(o))
21: ((o)(oo))
23: (((o)(o)))
30: (o(o)((o)))
33: ((o)(((o))))
35: (((o))(oo))
36: (oo(o)(o))
39: ((o)(o(o)))
42: (o(o)(oo))
46: (o((o)(o)))
47: (((o)((o))))
49: ((oo)(oo))
51: ((o)((oo)))
57: ((o)(ooo))
60: (oo(o)((o)))
61: ((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[#]]-1&]
CROSSREFS
The version for edge-height is A209638.
Square trees are A358577, counted by A358589, ordered A358590.
The version for leaves instead of height is A358578, counted by A185650.
These trees are counted by A358587, ordered A358588.
A000081 counts rooted trees, ordered A000108.
A034781 counts rooted trees by nodes and height.
A055277 counts rooted trees by leaves, ordered A001263.
Sequence in context: A289689 A316752 A358725 * A373995 A110473 A105441
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 25 2022
STATUS
approved