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A331490 Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).

%I A331490 #8 Feb 16 2025 08:33:59
%S A331490 8,16,28,32,56,64,76,98,112,128,152,172,196,212,224,256,266,304,343,
%T A331490 344,392,424,428,448,512,524,532,602,608,652,686,688,722,742,784,848,
%U A331490 856,896,908,931,1024,1048,1052,1064,1204,1216,1244,1304,1372,1376,1444
%N A331490 Matula-Goebel numbers of series-reduced rooted trees with more than two branches (of the root).
%C A331490 We say that a rooted tree is (topologically) series-reduced if no vertex has degree 2.
%C A331490 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 A331490 Also Matula-Goebel numbers of lone-child-avoiding rooted trees with more than two branches.
%H A331490 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 A331490 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Series-ReducedTree.html">Series-reduced tree.</a>
%H A331490 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 A331490 The sequence of all series-reduced rooted trees with more than two branches together with their Matula-Goebel numbers begins:
%e A331490     8: (ooo)
%e A331490    16: (oooo)
%e A331490    28: (oo(oo))
%e A331490    32: (ooooo)
%e A331490    56: (ooo(oo))
%e A331490    64: (oooooo)
%e A331490    76: (oo(ooo))
%e A331490    98: (o(oo)(oo))
%e A331490   112: (oooo(oo))
%e A331490   128: (ooooooo)
%e A331490   152: (ooo(ooo))
%e A331490   172: (oo(o(oo)))
%e A331490   196: (oo(oo)(oo))
%e A331490   212: (oo(oooo))
%e A331490   224: (ooooo(oo))
%e A331490   256: (oooooooo)
%e A331490   266: (o(oo)(ooo))
%e A331490   304: (oooo(ooo))
%e A331490   343: ((oo)(oo)(oo))
%e A331490   344: (ooo(o(oo)))
%t A331490 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A331490 srQ[n_]:=Or[n==1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
%t A331490 Select[Range[1000],PrimeOmega[#]>2&&srQ[#]&]
%Y A331490 These trees are counted by A331488.
%Y A331490 Unlabeled rooted trees are counted by A000081.
%Y A331490 Lone-child-avoiding rooted trees are counted by A001678.
%Y A331490 Topologically series-reduced rooted trees are counted by A001679.
%Y A331490 Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
%Y A331490 Matula-Goebel numbers of series-reduced rooted trees are A331489.
%Y A331490 Cf. A000014, A000669, A004250, A007097, A007821, A033942, A060313, A060356, A061775, A109082, A109129, A196050, A276625, A330943.
%K A331490 nonn
%O A331490 1,1
%A A331490 _Gus Wiseman_, Jan 20 2020