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Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.
6

%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