cp's OEIS Frontend

A331682 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.

%I A331682 #6 Feb 01 2020 07:09:01
%S A331682 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 A331682 32,33,34,35,37,38,40,41,43,44,47,48,51,52,53,55,56,58,59,60,62,64,66,
%U A331682 67,68,70,71,74,76,77,79,80,82,85,86,88,89,93,94,95,96,101
%N A331682 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 A331682 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 A331682 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 A331682 The sequence of all locally disjoint rooted semi-identity trees together with their Matula-Goebel numbers begins:
%e A331682    1: o
%e A331682    2: (o)
%e A331682    3: ((o))
%e A331682    4: (oo)
%e A331682    5: (((o)))
%e A331682    6: (o(o))
%e A331682    7: ((oo))
%e A331682    8: (ooo)
%e A331682   10: (o((o)))
%e A331682   11: ((((o))))
%e A331682   12: (oo(o))
%e A331682   13: ((o(o)))
%e A331682   14: (o(oo))
%e A331682   15: ((o)((o)))
%e A331682   16: (oooo)
%e A331682   17: (((oo)))
%e A331682   19: ((ooo))
%e A331682   20: (oo((o)))
%e A331682   22: (o(((o))))
%e A331682   24: (ooo(o))
%t A331682 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A331682 deQ[n_]:=n==1||PrimeQ[n]&&deQ[PrimePi[n]]||CoprimeQ@@primeMS[n]&&And@@deQ/@primeMS[n];
%t A331682 Select[Range[100],deQ]
%Y A331682 The non-semi identity tree case is A316494.
%Y A331682 The enumeration of these trees by vertices is A331783.
%Y A331682 Semi-identity trees are counted by A306200.
%Y A331682 Matula-Goebel numbers of semi-identity trees are A306202.
%Y A331682 Locally disjoint rooted trees are counted by A316473.
%Y A331682 Matula-Goebel numbers of locally disjoint rooted trees are A316495.
%Y A331682 Cf. A000081, A007097, A061775, A196050, A276625, A316470, A331681, A331683.
%K A331682 nonn
%O A331682 1,2
%A A331682 _Gus Wiseman_, Jan 27 2020