A330943 Matula-Goebel numbers of singleton-reduced rooted trees.

1, 2, 4, 6, 7, 8, 12, 13, 14, 16, 18, 19, 21, 24, 26, 28, 32, 34, 36, 37, 38, 39, 42, 43, 48, 49, 52, 53, 54, 56, 57, 61, 63, 64, 68, 72, 73, 74, 76, 78, 82, 84, 86, 89, 91, 96, 98, 101, 102, 104, 106, 107, 108, 111, 112, 114, 117, 119, 122, 126, 128, 129, 131

Offset: 1

Author

Gus Wiseman, Jan 13 2020

Keywords

• nonn

Comments

These trees are counted by A330951.

A rooted tree is singleton-reduced if no non-leaf node has all singleton branches, where a rooted tree is a singleton if its root has degree 1.

The Matula-Goebel number of a rooted tree is the product of primes of the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.

A prime index of n is a number m such that prime(m) divides n. A number belongs to this sequence iff it is 1 or its prime indices all belong to this sequence but are not all prime.

Examples

			The sequence of all singleton-reduced rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   7: ((oo))
   8: (ooo)
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  19: ((ooo))
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  34: (o((oo)))
  36: (oo(o)(o))
  37: ((oo(o)))
		

Crossrefs

The series-reduced case is A291636.

Unlabeled rooted trees are counted by A000081.

Numbers whose prime indices are not all prime are A330945.

Singleton-reduced rooted trees are counted by A330951.

Singleton-reduced phylogenetic trees are A000311.

The set S of numbers whose prime indices do not all belong to S is A324694.

Cf. A000669, A001678, A006450, A007097, A007821, A061775, A196050, A257994, A276625, A277098, A320628, A330944, A330948.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mgsingQ[n_]:=n==1||And@@mgsingQ/@primeMS[n]&&!And@@PrimeQ/@primeMS[n];
Select[Range[100],mgsingQ]