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A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

%I A331873 #5 Feb 03 2020 22:18:08
%S A331873 1,2,4,6,8,9,12,14,16,18,24,26,27,28,32,36,38,46,48,49,52,54,56,64,69,
%T A331873 72,74,76,81,86,92,96,98,104,106,108,112,122,128,138,144,148,152,161,
%U A331873 162,169,172,178,184,192,196,202,206,207,208,212,214,216,224,243
%N A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.
%C A331873 First differs from A331936 in having 69, the Matula-Goebel number of the tree ((o)((o)(o))).
%C A331873 A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
%C A331873 Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
%C A331873 The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
%C A331873 Consists of one, two, and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.
%H A331873 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 A331873 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 A331873 The sequence of all semi-lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
%e A331873    1: o
%e A331873    2: (o)
%e A331873    4: (oo)
%e A331873    6: (o(o))
%e A331873    8: (ooo)
%e A331873    9: ((o)(o))
%e A331873   12: (oo(o))
%e A331873   14: (o(oo))
%e A331873   16: (oooo)
%e A331873   18: (o(o)(o))
%e A331873   24: (ooo(o))
%e A331873   26: (o(o(o)))
%e A331873   27: ((o)(o)(o))
%e A331873   28: (oo(oo))
%e A331873   32: (ooooo)
%e A331873   36: (oo(o)(o))
%e A331873   38: (o(ooo))
%e A331873   46: (o((o)(o)))
%e A331873   48: (oooo(o))
%e A331873   49: ((oo)(oo))
%t A331873 msQ[n_]:=n==1||n==2||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
%t A331873 Select[Range[100],msQ]
%Y A331873 Not requiring lone-child-avoidance gives A316495.
%Y A331873 A superset of A320269.
%Y A331873 The semi-identity tree case is A331681.
%Y A331873 The non-semi version (i.e., not containing 2) is A331871.
%Y A331873 These trees counted by vertices are A331872.
%Y A331873 These trees counted by leaves are A331874.
%Y A331873 Not requiring local disjointness gives A331935.
%Y A331873 The identity tree case is A331937.
%Y A331873 Cf. A007097, A050381, A061775, A196050, A291636, A302696, A316473, A316696, A316697, A331680, A331682, A331683, A331687, A331934.
%K A331873 nonn
%O A331873 1,2
%A A331873 _Gus Wiseman_, Feb 02 2020