A331967 Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.

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

Offset: 1

Author

Gus Wiseman, Feb 06 2020

Keywords

• nonn

Comments

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.

Examples

			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))
		

Links

• Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
• Gus Wiseman, The first 30 lone-child-avoiding achiral rooted trees.

Crossrefs

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.

Programs

Mathematica

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

Formula

Intersection of A214577 (achiral) and A291636 (lone-child-avoiding).