A331682 - OEIS

%I #6 Feb 01 2020 07:09:01

%S 1,2,3,4,5,6,7,8,10,11,12,13,14,15,16,17,19,20,22,24,26,28,29,30,31,

%T 32,33,34,35,37,38,40,41,43,44,47,48,51,52,53,55,56,58,59,60,62,64,66,

%U 67,68,70,71,74,76,77,79,80,82,85,86,88,89,93,94,95,96,101

%N One and all numbers whose prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be coprime.

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%C Also Matula-Goebel numbers of locally disjoint rooted semi-identity trees. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches (of the root), which gives a bijective correspondence between positive integers and unlabeled rooted trees.

%e The sequence of all locally disjoint rooted semi-identity trees together with 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   19: ((ooo))

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

%e   22: (o(((o))))

%e   24: (ooo(o))

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

%t deQ[n_]:=n==1||PrimeQ[n]&&deQ[PrimePi[n]]||CoprimeQ@@primeMS[n]&&And@@deQ/@primeMS[n];

%t Select[Range[100],deQ]

%Y The non-semi identity tree case is A316494.

%Y The enumeration of these trees by vertices is A331783.

%Y Semi-identity trees are counted by A306200.

%Y Matula-Goebel numbers of semi-identity trees are A306202.

%Y Locally disjoint rooted trees are counted by A316473.

%Y Matula-Goebel numbers of locally disjoint rooted trees are A316495.

%Y Cf. A000081, A007097, A061775, A196050, A276625, A316470, A331681, A331683.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 27 2020