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A331992 Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees. 5

%I #5 Feb 06 2020 20:55:22

%S 1,2,4,8,9,16,27,32,49,64,81,128,243,256,343,361,512,529,729,1024,

%T 2048,2187,2401,2809,4096,6561,6859,8192,10609,12167,16384,16807,

%U 17161,19683,32768,51529,59049,65536,96721,117649,130321,131072,148877,175561,177147

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

%C 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 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, two, 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="/A331992/a331992.png">The first 36 semi-lone-child-avoiding achiral rooted trees.</a>

%F Intersection of A214577 (achiral) and A331935 (semi-lone-child-avoiding).

%e The sequence of all semi-lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:

%e 1: o

%e 2: (o)

%e 4: (oo)

%e 8: (ooo)

%e 9: ((o)(o))

%e 16: (oooo)

%e 27: ((o)(o)(o))

%e 32: (ooooo)

%e 49: ((oo)(oo))

%e 64: (oooooo)

%e 81: ((o)(o)(o)(o))

%e 128: (ooooooo)

%e 243: ((o)(o)(o)(o)(o))

%e 256: (oooooooo)

%e 343: ((oo)(oo)(oo))

%e 361: ((ooo)(ooo))

%e 512: (ooooooooo)

%e 529: (((o)(o))((o)(o)))

%e 729: ((o)(o)(o)(o)(o)(o))

%e 1024: (oooooooooo)

%t msQ[n_]:=n<=2||!PrimeQ[n]&&Length[FactorInteger[n]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];

%t Select[Range[10000],msQ]

%Y Except for two, a subset of A025475 (nonprime prime powers).

%Y Not requiring achirality gives A331935.

%Y The semi-achiral version is A331936.

%Y The fully-chiral version is A331963.

%Y The semi-chiral version is A331994.

%Y The non-semi version is counted by A331967.

%Y The enumeration of these trees by vertices is A331991.

%Y Achiral rooted trees are counted by A003238.

%Y MG-numbers of achiral rooted trees are A214577.

%Y Cf. A001678, A007097, A050381, A061775, A167865, A196050, A276625, A280996, A291441, A291636, A320230, A320269, A331912, A331933, A331965.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 06 2020

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)