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A331992
Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees.
5
1, 2, 4, 8, 9, 16, 27, 32, 49, 64, 81, 128, 243, 256, 343, 361, 512, 529, 729, 1024, 2048, 2187, 2401, 2809, 4096, 6561, 6859, 8192, 10609, 12167, 16384, 16807, 17161, 19683, 32768, 51529, 59049, 65536, 96721, 117649, 130321, 131072, 148877, 175561, 177147
OFFSET
1,2
COMMENTS
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.
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, two, and all numbers of the form prime(j)^k where k > 1 and j is already in the sequence.
FORMULA
Intersection of A214577 (achiral) and A331935 (semi-lone-child-avoiding).
EXAMPLE
The sequence of all semi-lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
4: (oo)
8: (ooo)
9: ((o)(o))
16: (oooo)
27: ((o)(o)(o))
32: (ooooo)
49: ((oo)(oo))
64: (oooooo)
81: ((o)(o)(o)(o))
128: (ooooooo)
243: ((o)(o)(o)(o)(o))
256: (oooooooo)
343: ((oo)(oo)(oo))
361: ((ooo)(ooo))
512: (ooooooooo)
529: (((o)(o))((o)(o)))
729: ((o)(o)(o)(o)(o)(o))
1024: (oooooooooo)
MATHEMATICA
msQ[n_]:=n<=2||!PrimeQ[n]&&Length[FactorInteger[n]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[10000], msQ]
CROSSREFS
Except for two, a subset of A025475 (nonprime prime powers).
Not requiring achirality gives A331935.
The semi-achiral version is A331936.
The fully-chiral version is A331963.
The semi-chiral version is A331994.
The non-semi version is counted by A331967.
The enumeration of these trees by vertices is A331991.
Achiral rooted trees are counted by A003238.
MG-numbers of achiral rooted trees are A214577.
Sequence in context: A046678 A330606 A046680 * A113570 A065391 A161792
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 06 2020
STATUS
approved