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%I #8 Jul 05 2018 02:30:51
%S 1,2,3,5,6,10,11,13,15,22,26,29,30,31,33,41,47,55,58,62,66,79,82,93,
%T 94,101,109,110,113,123,127,137,141,143,145,155,158,165,179,186,202,
%U 205,211,218,226,246,254,257,271,274,282,286,290,293,310,317,327,330
%N Matula-Goebel numbers of locally disjoint rooted identity trees, meaning no branch overlaps 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 is in the sequence iff either it is equal to 1, it is a prime number whose prime index already belongs to the sequence, or its prime indices are pairwise coprime, distinct, and already belong to the sequence.
%e The sequence of all locally disjoint rooted identity trees preceded by their Matula-Goebel numbers begins:
%e 1: o
%e 2: (o)
%e 3: ((o))
%e 5: (((o)))
%e 6: (o(o))
%e 10: (o((o)))
%e 11: ((((o))))
%e 13: ((o(o)))
%e 15: ((o)((o)))
%e 22: (o(((o))))
%e 26: (o(o(o)))
%e 29: ((o((o))))
%e 30: (o(o)((o)))
%e 31: (((((o)))))
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[1000],Or[#==1,And[SquareFreeQ[#],Or[PrimeQ[#],CoprimeQ@@primeMS[#]],And@@#0/@primeMS[#]]]&]
%Y Cf. A000081, A004111, A007097, A276625, A277098, A302696, A303362, A304713, A316467, A316471, A316474, A316495.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jul 04 2018
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