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%I #7 Jul 04 2018 20:24:57
%S 1,2,3,5,11,15,31,33,47,55,93,127,137,141,155,165,211,257,341,381,411,
%T 465,487,633,635,709,771,773,811,907,977,1023,1055,1285,1297,1397,
%U 1457,1461,1507,1621,1705,1905,2127,2293,2319,2321,2433,2621,2721,2833,2931
%N Matula-Goebel numbers of locally stable rooted identity trees, meaning no branch is a subset of any other branch of the same root.
%C A prime index of n is a number m such that prime(m) divides n. A number belongs to this sequence iff it is squarefree, its distinct prime indices are pairwise indivisible, and its prime indices also belong to this sequence.
%e 165 = prime(2)*prime(3)*prime(5) belongs to the sequence because it is squarefree, the indices {2,3,5} are pairwise indivisible, and each of them already belongs to the sequence.
%e Sequence of locally stable rooted identity trees preceded by their Matula-Goebel numbers begins:
%e 1: o
%e 2: (o)
%e 3: ((o))
%e 5: (((o)))
%e 11: ((((o))))
%e 15: ((o)((o)))
%e 31: (((((o)))))
%e 33: ((o)(((o))))
%e 47: (((o)((o))))
%e 55: (((o))(((o))))
%e 93: ((o)((((o)))))
%e 127: ((((((o))))))
%e 137: (((o)(((o)))))
%e 141: ((o)((o)((o))))
%e 155: (((o))((((o)))))
%e 165: ((o)((o))(((o))))
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t ain[n_]:=And[Select[Tuples[primeMS[n],2],UnsameQ@@#&&Divisible@@#&]=={},SquareFreeQ[n],And@@ain/@primeMS[n]];
%t Select[Range[100],ain]
%Y Cf. A000081, A004111, A007097, A276625, A277098, A285572, A285573, A302796, A303362, A304713, A316468, A316469, A316471, A316474, A316476, A316494.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jul 04 2018
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