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

%I A291636 #17 Jun 25 2021 23:41:16
%S A291636 1,4,8,14,16,28,32,38,49,56,64,76,86,98,106,112,128,133,152,172,196,
%T A291636 212,214,224,256,262,266,301,304,326,343,344,361,371,392,424,428,448,
%U A291636 454,512,524,526,532,602,608,622,652,686,688,722,742,749,766,784,817
%N A291636 Matula-Goebel numbers of lone-child-avoiding rooted trees.
%C A291636 We say that a rooted tree is lone-child-avoiding if no vertex has exactly one child.
%C A291636 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.
%C A291636 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.
%H A291636 David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], (30-June-2014).
%H A291636 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>
%H A291636 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%e A291636 The sequence of all lone-child-avoiding rooted trees together with their Matula-Goebel numbers begins:
%e A291636     1: o
%e A291636     4: (oo)
%e A291636     8: (ooo)
%e A291636    14: (o(oo))
%e A291636    16: (oooo)
%e A291636    28: (oo(oo))
%e A291636    32: (ooooo)
%e A291636    38: (o(ooo))
%e A291636    49: ((oo)(oo))
%e A291636    56: (ooo(oo))
%e A291636    64: (oooooo)
%e A291636    76: (oo(ooo))
%e A291636    86: (o(o(oo)))
%e A291636    98: (o(oo)(oo))
%e A291636   106: (o(oooo))
%e A291636   112: (oooo(oo))
%e A291636   128: (ooooooo)
%e A291636   133: ((oo)(ooo))
%e A291636   152: (ooo(ooo))
%e A291636   172: (oo(o(oo)))
%t A291636 nn=2000;
%t A291636 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A291636 srQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
%t A291636 Select[Range[nn],srQ]
%Y A291636 These trees are counted by A001678.
%Y A291636 The case with more than two branches is A331490.
%Y A291636 Unlabeled rooted trees are counted by A000081.
%Y A291636 Topologically series-reduced rooted trees are counted by A001679.
%Y A291636 Labeled lone-child-avoiding rooted trees are counted by A060356.
%Y A291636 Labeled lone-child-avoiding unrooted trees are counted by A108919.
%Y A291636 MG numbers of singleton-reduced rooted trees are A330943.
%Y A291636 MG numbers of topologically series-reduced rooted trees are A331489.
%Y A291636 Cf. A007097, A061775, A109082, A109129, A111299, A196050, A198518, A276625, A291441, A291442, A331488.
%K A291636 nonn
%O A291636 1,2
%A A291636 _Gus Wiseman_, Aug 28 2017
%E A291636 Updated with corrected terminology by _Gus Wiseman_, Jan 20 2020