cp's OEIS Frontend

A331967 Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.

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