A316495 - OEIS

%I #6 Jul 05 2018 02:30:59

%S 1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,18,19,20,22,24,26,28,29,30,

%T 31,32,33,34,35,36,37,38,40,41,43,44,45,47,48,50,51,52,53,54,55,56,58,

%U 59,60,61,62,64,66,67,68,70,71,72,74,75,76,77,79,80,82,85

%N Matula-Goebel numbers of locally disjoint unlabeled rooted trees, meaning no branch overlaps any other (unequal) 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 distinct prime indices are pairwise coprime and already belong to the sequence.

%e The sequence of all locally disjoint rooted trees preceded by their Matula-Goebel numbers begins:

%e    1: o

%e    2: (o)

%e    3: ((o))

%e    4: (oo)

%e    5: (((o)))

%e    6: (o(o))

%e    7: ((oo))

%e    8: (ooo)

%e   10: (o((o)))

%e   11: ((((o))))

%e   12: (oo(o))

%e   13: ((o(o)))

%e   14: (o(oo))

%e   15: ((o)((o)))

%e   16: (oooo)

%e   17: (((oo)))

%e   18: (o(o)(o))

%e   19: ((ooo))

%e   20: (oo((o)))

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t go[n_]:=Or[n==1,And[Or[PrimeQ[n],CoprimeQ@@Union[primeMS[n]]],And@@go/@primeMS[n]]];

%t Select[Range[100],go]

%Y Cf. A000081, A007097, A302696, A303362, A304713, A316468, A316470, A316473, A316475, A316494.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 04 2018