

A331992


MatulaGoebel numbers of semilonechildavoiding 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
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OFFSET

1,2


COMMENTS

A rooted tree is semilonechildavoiding 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 MatulaGoebel number of a rooted tree is the product of primes indexed by the MatulaGoebel 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.


LINKS

Table of n, a(n) for n=1..45.
Gus Wiseman, Sequences counting seriesreduced and lonechildavoiding trees by number of vertices.
Gus Wiseman, The first 36 semilonechildavoiding achiral rooted trees.


FORMULA

Intersection of A214577 (achiral) and A331935 (semilonechildavoiding).


EXAMPLE

The sequence of all semilonechildavoiding achiral rooted trees together with their MatulaGoebel 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 semiachiral version is A331936.
The fullychiral version is A331963.
The semichiral version is A331994.
The nonsemi version is counted by A331967.
The enumeration of these trees by vertices is A331991.
Achiral rooted trees are counted by A003238.
MGnumbers of achiral rooted trees are A214577.
Cf. A001678, A007097, A050381, A061775, A167865, A196050, A276625, A280996, A291441, A291636, A320230, A320269, A331912, A331933, A331965.
Sequence in context: A046678 A330606 A046680 * A113570 A065391 A161792
Adjacent sequences: A331989 A331990 A331991 * A331993 A331994 A331995


KEYWORD

nonn


AUTHOR

Gus Wiseman, Feb 06 2020


STATUS

approved



