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