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A331937 a(1) = 1; a(2) = 2; a(n + 1) = 2 * prime(a(n)).

%I A331937 #7 Feb 10 2020 15:20:35
%S A331937 1,2,6,26,202,2462,43954,1063462,33076174,1270908802,58596709306,
%T A331937 3170266564862,197764800466826,14024066291995502,1117378164606478094
%N A331937 a(1) = 1; a(2) = 2; a(n + 1) = 2 * prime(a(n)).
%C A331937 Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted identity trees. A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex. It is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. In an identity tree, the branches of any given vertex are all distinct. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
%F A331937 Intersection of A276625 (identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding).
%e A331937 The sequence of terms together with their associated trees begins:
%e A331937      1: o
%e A331937      2: (o)
%e A331937      6: (o(o))
%e A331937     26: (o(o(o)))
%e A331937    202: (o(o(o(o))))
%e A331937   2462: (o(o(o(o(o)))))
%t A331937 msiQ[n_]:=n==1||n==2||!PrimeQ[n]&&SquareFreeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msiQ/@PrimePi/@First/@FactorInteger[n];
%t A331937 Select[Range[1000],msiQ]
%Y A331937 The semi-identity tree version is A331681.
%Y A331937 Not requiring an identity tree gives A331873.
%Y A331937 Not requiring local disjointness gives A331963.
%Y A331937 Not requiring lone-child-avoidance gives A316494.
%Y A331937 MG-numbers of semi-lone-child-avoiding rooted trees are A331935.
%Y A331937 Cf. A007097, A061775, A276625, A316471, A316495, A316694, A331679, A331683, A331686, A331872, A331934, A331936, A331964, A331965.
%K A331937 nonn,more
%O A331937 1,2
%A A331937 _Gus Wiseman_, Feb 07 2020
%E A331937 a(14)-a(15) from _Giovanni Resta_, Feb 10 2020