%I #4 Feb 06 2020 20:55:05
%S 1,4,8,16,32,49,64,128,256,343,361,512,1024,2048,2401,2809,4096,6859,
%T 8192,16384,16807,17161,32768,51529,65536,96721,117649,130321,131072,
%U 148877,262144,516961,524288,823543,1048576,2097152,2248091,2476099,2621161,4194304
%N Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.
%C Lone-child-avoiding means there are no unary branchings.
%C In an achiral rooted tree, the branches of any given vertex are all equal.
%C 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 Consists of one and all numbers of the form prime(j)^k where k > 1 and j is already in the sequence.
%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>
%H Gus Wiseman, <a href="/A331967/a331967.png">The first 30 lone-child-avoiding achiral rooted trees.</a>
%F Intersection of A214577 (achiral) and A291636 (lone-child-avoiding).
%e The sequence of all lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
%e 1: o
%e 4: (oo)
%e 8: (ooo)
%e 16: (oooo)
%e 32: (ooooo)
%e 49: ((oo)(oo))
%e 64: (oooooo)
%e 128: (ooooooo)
%e 256: (oooooooo)
%e 343: ((oo)(oo)(oo))
%e 361: ((ooo)(ooo))
%e 512: (ooooooooo)
%e 1024: (oooooooooo)
%e 2048: (ooooooooooo)
%e 2401: ((oo)(oo)(oo)(oo))
%e 2809: ((oooo)(oooo))
%e 4096: (oooooooooooo)
%e 6859: ((ooo)(ooo)(ooo))
%e 8192: (ooooooooooooo)
%e 16384: (oooooooooooooo)
%e 16807: ((oo)(oo)(oo)(oo)(oo))
%e 17161: ((ooooo)(ooooo))
%e 32768: (ooooooooooooooo)
%e 51529: (((oo)(oo))((oo)(oo)))
%e 65536: (oooooooooooooooo)
%e 96721: ((oooooo)(oooooo))
%t msQ[n_]:=n==1||!PrimeQ[n]&&PrimePowerQ[n]&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
%t Select[Range[10000],msQ]
%Y A subset of A025475 (nonprime prime powers).
%Y The enumeration of these trees by vertices is A167865.
%Y Not requiring lone-child-avoidance gives A214577.
%Y The semi-achiral version is A320269.
%Y The semi-lone-child-avoiding version is A331992.
%Y Achiral rooted trees are counted by A003238.
%Y MG-numbers of planted achiral rooted trees are A280996.
%Y MG-numbers of lone-child-avoiding rooted trees are A291636.
%Y Cf. A001678, A007097, A061775, A196050, A276625, A291441, A320230, A320268, A331912, A331936, A331963, A331965, A331966, A331991.
%K nonn
%O 1,2
%A _Gus Wiseman_, Feb 06 2020