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A358725
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Matula-Goebel numbers of rooted trees with a greater number of internal (non-leaf) vertices than edge-height.
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5
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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)))
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MATHEMATICA
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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&]
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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