A331490 - OEIS

%I #7 Jan 20 2020 21:44:28

%S 8,16,28,32,56,64,76,98,112,128,152,172,196,212,224,256,266,304,343,

%T 344,392,424,428,448,512,524,532,602,608,652,686,688,722,742,784,848,

%U 856,896,908,931,1024,1048,1052,1064,1204,1216,1244,1304,1372,1376,1444

%N Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).

%C We say that a rooted tree is (topologically) series-reduced if no vertex 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.

%C Also Matula-Goebel numbers of lone-child-avoiding rooted trees with more than two branches.

%H 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 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 series-reduced rooted trees with more than two branches together with their Matula-Goebel numbers begins:

%e     8: (ooo)

%e    16: (oooo)

%e    28: (oo(oo))

%e    32: (ooooo)

%e    56: (ooo(oo))

%e    64: (oooooo)

%e    76: (oo(ooo))

%e    98: (o(oo)(oo))

%e   112: (oooo(oo))

%e   128: (ooooooo)

%e   152: (ooo(ooo))

%e   172: (oo(o(oo)))

%e   196: (oo(oo)(oo))

%e   212: (oo(oooo))

%e   224: (ooooo(oo))

%e   256: (oooooooo)

%e   266: (o(oo)(ooo))

%e   304: (oooo(ooo))

%e   343: ((oo)(oo)(oo))

%e   344: (ooo(o(oo)))

%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[1000],PrimeOmega[#]>2&&srQ[#]&]

%Y These trees are counted by A331488.

%Y Unlabeled rooted trees are counted by A000081.

%Y Lone-child-avoiding rooted trees are counted by A001678.

%Y Topologically series-reduced rooted trees are counted by A001679.

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

%Y Matula-Goebel numbers of series-reduced rooted trees are A331489.

%Y Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jan 20 2020