A291636 Matula-Goebel numbers of lone-child-avoiding rooted trees.

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 133, 152, 172, 196, 212, 214, 224, 256, 262, 266, 301, 304, 326, 343, 344, 361, 371, 392, 424, 428, 448, 454, 512, 524, 526, 532, 602, 608, 622, 652, 686, 688, 722, 742, 749, 766, 784, 817

Offset

1,2

Comments

We say that a rooted tree is lone-child-avoiding if no vertex has exactly one child.

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

An alternative definition: n is in the sequence iff n is 1 or the product of two or more not necessarily distinct prime numbers whose prime indices already belong to the sequence. For example, 14 is in the sequence because 14 = prime(1) * prime(4) and 1 and 4 both already belong to the sequence.

Links

Table of n, a(n) for n=1..55.

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).

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

Index entries for sequences related to Matula-Goebel numbers

Example

The sequence of all lone-child-avoiding rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  133: ((oo)(ooo))
  152: (ooo(ooo))
  172: (oo(o(oo)))

Mathematica

nn=2000;
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
srQ[n_]:=Or[n===1, With[{m=primeMS[n]}, And[Length[m]>1, And@@srQ/@m]]];
Select[Range[nn], srQ]

Crossrefs

These trees are counted by A001678.

The case with more than two branches is A331490.

Unlabeled rooted trees are counted by A000081.

Topologically series-reduced rooted trees are counted by A001679.

Labeled lone-child-avoiding rooted trees are counted by A060356.

Labeled lone-child-avoiding unrooted trees are counted by A108919.

MG numbers of singleton-reduced rooted trees are A330943.

MG numbers of topologically series-reduced rooted trees are A331489.

Cf. A007097, A061775, A109082, A109129, A111299, A196050, A198518, A276625, A291441, A291442, A331488.

Sequence in context: A312335 A312336 A312337 * A320269 A331871 A331965

Adjacent sequences: A291633 A291634 A291635 * A291637 A291638 A291639

Keyword

nonn

Author

Gus Wiseman, Aug 28 2017

Extensions

Updated with corrected terminology by Gus Wiseman, Jan 20 2020

Status

approved