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A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

%I A331936 #15 Feb 05 2020 23:54:21
%S A331936 1,2,4,6,8,9,12,14,16,18,24,26,27,28,32,36,38,46,48,49,52,54,56,64,72,
%T A331936 74,76,81,86,92,96,98,104,106,108,112,122,128,144,148,152,162,169,172,
%U A331936 178,184,192,196,202,206,208,212,214,216,224,243,244,256,262,288
%N A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).
%C A331936 First differs from A331873 in lacking 69, the Matula-Goebel number of the tree ((o)((o)(o))).
%C A331936 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 A331936 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 A331936 Consists of 1, 2, and all numbers equal to a power of 2 (other than 1) times a power of prime(j) for some j > 1 already in the sequence.
%H A331936 David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], (30-June-2014).
%H A331936 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>
%F A331936 Intersection of A320230 and A331935.
%e A331936 The sequence of rooted trees ranked by this sequence together with their Matula-Goebel numbers begins:
%e A331936    1: o
%e A331936    2: (o)
%e A331936    4: (oo)
%e A331936    6: (o(o))
%e A331936    8: (ooo)
%e A331936    9: ((o)(o))
%e A331936   12: (oo(o))
%e A331936   14: (o(oo))
%e A331936   16: (oooo)
%e A331936   18: (o(o)(o))
%e A331936   24: (ooo(o))
%e A331936   26: (o(o(o)))
%e A331936   27: ((o)(o)(o))
%e A331936   28: (oo(oo))
%e A331936   32: (ooooo)
%e A331936   36: (oo(o)(o))
%e A331936   38: (o(ooo))
%e A331936   46: (o((o)(o)))
%e A331936   48: (oooo(o))
%e A331936   49: ((oo)(oo))
%e A331936 The sequence of terms together with their prime indices begins:
%e A331936     1: {}              52: {1,1,6}            152: {1,1,1,8}
%e A331936     2: {1}             54: {1,2,2,2}          162: {1,2,2,2,2}
%e A331936     4: {1,1}           56: {1,1,1,4}          169: {6,6}
%e A331936     6: {1,2}           64: {1,1,1,1,1,1}      172: {1,1,14}
%e A331936     8: {1,1,1}         72: {1,1,1,2,2}        178: {1,24}
%e A331936     9: {2,2}           74: {1,12}             184: {1,1,1,9}
%e A331936    12: {1,1,2}         76: {1,1,8}            192: {1,1,1,1,1,1,2}
%e A331936    14: {1,4}           81: {2,2,2,2}          196: {1,1,4,4}
%e A331936    16: {1,1,1,1}       86: {1,14}             202: {1,26}
%e A331936    18: {1,2,2}         92: {1,1,9}            206: {1,27}
%e A331936    24: {1,1,1,2}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
%e A331936    26: {1,6}           98: {1,4,4}            212: {1,1,16}
%e A331936    27: {2,2,2}        104: {1,1,1,6}          214: {1,28}
%e A331936    28: {1,1,4}        106: {1,16}             216: {1,1,1,2,2,2}
%e A331936    32: {1,1,1,1,1}    108: {1,1,2,2,2}        224: {1,1,1,1,1,4}
%e A331936    36: {1,1,2,2}      112: {1,1,1,1,4}        243: {2,2,2,2,2}
%e A331936    38: {1,8}          122: {1,18}             244: {1,1,18}
%e A331936    46: {1,9}          128: {1,1,1,1,1,1,1}    256: {1,1,1,1,1,1,1,1}
%e A331936    48: {1,1,1,1,2}    144: {1,1,1,1,2,2}      262: {1,32}
%e A331936    49: {4,4}          148: {1,1,12}           288: {1,1,1,1,1,2,2}
%t A331936 msQ[n_]:=n<=2||!PrimeQ[n]&&Length[DeleteCases[FactorInteger[n],{2,_}]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
%t A331936 Select[Range[100],msQ]
%Y A331936 A superset of A000079.
%Y A331936 The non-lone-child-avoiding version is A320230.
%Y A331936 The non-semi version is A320269.
%Y A331936 These trees are counted by A331933.
%Y A331936 Not requiring semi-achirality gives A331935.
%Y A331936 The fully-achiral case is A331992.
%Y A331936 Achiral trees are counted by A003238.
%Y A331936 Numbers with at most one distinct odd prime factor are A070776.
%Y A331936 Matula-Goebel numbers of achiral rooted trees are A214577.
%Y A331936 Matula-Goebel numbers of semi-identity trees are A306202.
%Y A331936 Numbers S with at most one distinct prime index in S are A331912.
%Y A331936 Cf. A001678, A007097, A050381, A061775, A196050, A291636, A331784, A331873, A331914, A331934, A331965, A331967, A331991.
%K A331936 nonn
%O A331936 1,2
%A A331936 _Gus Wiseman_, Feb 03 2020