A331992 Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees.

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

Author

Gus Wiseman, Feb 06 2020

Keywords

• nonn

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.

Examples

			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)
		

Links

• 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.

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.

Programs

Mathematica

msQ[n_]:=n<=2||!PrimeQ[n]&&Length[FactorInteger[n]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[10000],msQ]

Formula

Intersection of A214577 (achiral) and A331935 (semi-lone-child-avoiding).