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