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A331967
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Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.
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6
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1, 4, 8, 16, 32, 49, 64, 128, 256, 343, 361, 512, 1024, 2048, 2401, 2809, 4096, 6859, 8192, 16384, 16807, 17161, 32768, 51529, 65536, 96721, 117649, 130321, 131072, 148877, 262144, 516961, 524288, 823543, 1048576, 2097152, 2248091, 2476099, 2621161, 4194304
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OFFSET
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1,2
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COMMENTS
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Lone-child-avoiding means there are no unary branchings.
In an achiral rooted tree, the branches of any given vertex are all equal.
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.
Consists of one and all numbers of the form prime(j)^k where k > 1 and j is already in the sequence.
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LINKS
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FORMULA
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EXAMPLE
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The sequence of all lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
1: o
4: (oo)
8: (ooo)
16: (oooo)
32: (ooooo)
49: ((oo)(oo))
64: (oooooo)
128: (ooooooo)
256: (oooooooo)
343: ((oo)(oo)(oo))
361: ((ooo)(ooo))
512: (ooooooooo)
1024: (oooooooooo)
2048: (ooooooooooo)
2401: ((oo)(oo)(oo)(oo))
2809: ((oooo)(oooo))
4096: (oooooooooooo)
6859: ((ooo)(ooo)(ooo))
8192: (ooooooooooooo)
16384: (oooooooooooooo)
16807: ((oo)(oo)(oo)(oo)(oo))
17161: ((ooooo)(ooooo))
32768: (ooooooooooooooo)
51529: (((oo)(oo))((oo)(oo)))
65536: (oooooooooooooooo)
96721: ((oooooo)(oooooo))
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MATHEMATICA
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msQ[n_]:=n==1||!PrimeQ[n]&&PrimePowerQ[n]&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[10000], msQ]
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CROSSREFS
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A subset of A025475 (nonprime prime powers).
The enumeration of these trees by vertices is A167865.
Not requiring lone-child-avoidance gives A214577.
The semi-achiral version is A320269.
The semi-lone-child-avoiding version is A331992.
Achiral rooted trees are counted by A003238.
MG-numbers of planted achiral rooted trees are A280996.
MG-numbers of lone-child-avoiding rooted trees are A291636.
Cf. A001678, A007097, A061775, A196050, A276625, A291441, A320230, A320268, A331912, A331936, A331963, A331965, A331966, A331991.
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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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