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A331992 Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees.

%I A331992 #5 Feb 06 2020 20:55:22
%S A331992 1,2,4,8,9,16,27,32,49,64,81,128,243,256,343,361,512,529,729,1024,
%T A331992 2048,2187,2401,2809,4096,6561,6859,8192,10609,12167,16384,16807,
%U A331992 17161,19683,32768,51529,59049,65536,96721,117649,130321,131072,148877,175561,177147
%N A331992 Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees.
%C A331992 A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.
%C A331992 In an achiral rooted tree, the branches of any given vertex are all equal.
%C A331992 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 A331992 Consists of one, two, and all numbers of the form prime(j)^k where k > 1 and j is already in the sequence.
%H A331992 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 A331992 Gus Wiseman, <a href="/A331992/a331992.png">The first 36 semi-lone-child-avoiding achiral rooted trees.</a>
%F A331992 Intersection of A214577 (achiral) and A331935 (semi-lone-child-avoiding).
%e A331992 The sequence of all semi-lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
%e A331992      1: o
%e A331992      2: (o)
%e A331992      4: (oo)
%e A331992      8: (ooo)
%e A331992      9: ((o)(o))
%e A331992     16: (oooo)
%e A331992     27: ((o)(o)(o))
%e A331992     32: (ooooo)
%e A331992     49: ((oo)(oo))
%e A331992     64: (oooooo)
%e A331992     81: ((o)(o)(o)(o))
%e A331992    128: (ooooooo)
%e A331992    243: ((o)(o)(o)(o)(o))
%e A331992    256: (oooooooo)
%e A331992    343: ((oo)(oo)(oo))
%e A331992    361: ((ooo)(ooo))
%e A331992    512: (ooooooooo)
%e A331992    529: (((o)(o))((o)(o)))
%e A331992    729: ((o)(o)(o)(o)(o)(o))
%e A331992   1024: (oooooooooo)
%t A331992 msQ[n_]:=n<=2||!PrimeQ[n]&&Length[FactorInteger[n]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
%t A331992 Select[Range[10000],msQ]
%Y A331992 Except for two, a subset of A025475 (nonprime prime powers).
%Y A331992 Not requiring achirality gives A331935.
%Y A331992 The semi-achiral version is A331936.
%Y A331992 The fully-chiral version is A331963.
%Y A331992 The semi-chiral version is A331994.
%Y A331992 The non-semi version is counted by A331967.
%Y A331992 The enumeration of these trees by vertices is A331991.
%Y A331992 Achiral rooted trees are counted by A003238.
%Y A331992 MG-numbers of achiral rooted trees are A214577.
%Y A331992 Cf. A001678, A007097, A050381, A061775, A167865, A196050, A276625, A280996, A291441, A291636, A320230, A320269, A331912, A331933, A331965.
%K A331992 nonn
%O A331992 1,2
%A A331992 _Gus Wiseman_, Feb 06 2020