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%I #4 Jan 20 2020 21:44:18
%S 1,2,7,8,16,19,28,32,43,53,56,64,76,98,107,112,128,131,152,163,172,
%T 196,212,224,227,256,263,266,304,311,343,344,383,392,424,428,443,448,
%U 512,521,524,532,577,602,608,613,652,686,688,719,722,742,751,784,848,856
%N Matula-Goebel numbers of topologically series-reduced rooted trees.
%C We say that a rooted tree is topologically series-reduced if no vertex (including the root) has degree 2.
%C 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.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Series-ReducedTree.html">Series-reduced tree.</a>
%H 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>
%e The sequence of all topologically series-reduced rooted trees together with their Matula-Goebel numbers begins:
%e 1: o
%e 2: (o)
%e 7: ((oo))
%e 8: (ooo)
%e 16: (oooo)
%e 19: ((ooo))
%e 28: (oo(oo))
%e 32: (ooooo)
%e 43: ((o(oo)))
%e 53: ((oooo))
%e 56: (ooo(oo))
%e 64: (oooooo)
%e 76: (oo(ooo))
%e 98: (o(oo)(oo))
%e 107: ((oo(oo)))
%e 112: (oooo(oo))
%e 128: (ooooooo)
%e 131: ((ooooo))
%e 152: (ooo(ooo))
%e 163: ((o(ooo)))
%t nn=1000;
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t srQ[n_]:=Or[n==1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
%t Select[Range[nn],PrimeOmega[#]!=2&&And@@srQ/@primeMS[#]&]
%Y Unlabeled rooted trees are counted by A000081.
%Y Topologically series-reduced trees are counted by A000014.
%Y Topologically series-reduced rooted trees are counted by A001679.
%Y Labeled topologically series-reduced trees are counted by A005512.
%Y Labeled topologically series-reduced rooted trees are counted by A060313.
%Y Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
%Y Cf. A000669, A001678, A007097, A007821, A060356, A061775, A109082, A109129, A196050, A254382, A276625, A330943, A331490.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 20 2020
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