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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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..45.

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Gus Wiseman, The first 36 semi-lone-child-avoiding achiral rooted trees.

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.

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

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Last modified April 1 05:33 EDT 2020. Contains 333155 sequences. (Running on oeis4.)