cp's OEIS Frontend

A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).

%I A320269 #8 Feb 07 2020 09:05:11
%S A320269 1,4,8,14,16,28,32,38,49,56,64,76,86,98,106,112,128,152,172,196,212,
%T A320269 214,224,256,262,304,326,343,344,361,392,424,428,448,454,512,524,526,
%U A320269 608,622,652,686,688,722,766,784,848,856,886,896,908,1024,1042,1048,1052
%N A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).
%C A320269 First differs from A331871 in lacking 1589.
%C A320269 Lone-child-avoiding means there are no unary branchings.
%C A320269 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.
%H A320269 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>
%e A320269 The sequence of rooted trees together with their Matula-Goebel numbers begins:
%e A320269     1: o
%e A320269     4: (oo)
%e A320269     8: (ooo)
%e A320269    14: (o(oo))
%e A320269    16: (oooo)
%e A320269    28: (oo(oo))
%e A320269    32: (ooooo)
%e A320269    38: (o(ooo))
%e A320269    49: ((oo)(oo))
%e A320269    56: (ooo(oo))
%e A320269    64: (oooooo)
%e A320269    76: (oo(ooo))
%e A320269    86: (o(o(oo)))
%e A320269    98: (o(oo)(oo))
%e A320269   106: (o(oooo))
%e A320269   112: (oooo(oo))
%e A320269   128: (ooooooo)
%e A320269   152: (ooo(ooo))
%e A320269   172: (oo(o(oo)))
%e A320269   196: (oo(oo)(oo))
%t A320269 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
%t A320269 hmakQ[n_]:=And[!PrimeQ[n],SameQ@@DeleteCases[primeMS[n],1],And@@hmakQ/@primeMS[n]];Select[Range[1000],hmakQ[#]&]
%Y A320269 Cf. A002541, A070776, A167865, A214577, A317710, A320222, A320226.
%Y A320269 The same-tree version is A291441.
%Y A320269 Not requiring lone-child-avoidance gives A320230.
%Y A320269 The enumeration of these trees by vertices is A320268.
%Y A320269 The semi-lone-child-avoiding version is A331936.
%Y A320269 If the non-leaf branches are all different instead of equal we get A331965.
%Y A320269 The fully-achiral case is A331967.
%Y A320269 Achiral rooted trees are counted by A003238.
%Y A320269 MG-numbers of lone-child-avoiding rooted trees are A291636.
%Y A320269 Cf. A061775, A196050, A276625, A280996, A306202, A331912, A331916, A331966.
%K A320269 nonn
%O A320269 1,2
%A A320269 _Gus Wiseman_, Oct 08 2018
%E A320269 Updated with corrected terminology by _Gus Wiseman_, Feb 06 2020