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

1,2

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.

Links

Table of n, a(n) for n=1..63.

Example

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)))

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]

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.

Sequence in context: A228370 A186112 A029453 * A352089 A358458 A014855

Adjacent sequences: A330940 A330941 A330942 * A330944 A330945 A330946

Keyword

nonn

Author

Gus Wiseman, Jan 13 2020

Status

approved