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A331963 Matula-Goebel numbers of semi-lone-child-avoiding rooted identity trees.

%I A331963 #12 Feb 15 2020 11:04:39
%S A331963 1,2,6,26,39,78,202,303,334,501,606,794,1002,1191,1313,2171,2382,2462,
%T A331963 2626,3693,3939,3998,4342,4486,5161,5997,6513,6729,7162,7386,7878,
%U A331963 8914,10322,10743,11994,12178,13026,13371,13458,15483,15866,16003,16867,18267,19286
%N A331963 Matula-Goebel numbers of semi-lone-child-avoiding rooted identity trees.
%C A331963 A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. It is an identity tree if the branches under any given vertex are all distinct.
%C A331963 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.
%C A331963 Consists of one, two, and all nonprime squarefree numbers whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.
%H A331963 Gus Wiseman, <a href="/A331963/a331963.png">The first 63 semi-lone-child-avoiding rooted identity trees.</a>
%F A331963 Intersection of A276625 (identity trees) and A331935 (semi-lone-child-avoiding).
%e A331963 The sequence of all semi-lone-child-avoiding rooted identity trees together with their Matula-Goebel numbers begins:
%e A331963     1: o
%e A331963     2: (o)
%e A331963     6: (o(o))
%e A331963    26: (o(o(o)))
%e A331963    39: ((o)(o(o)))
%e A331963    78: (o(o)(o(o)))
%e A331963   202: (o(o(o(o))))
%e A331963   303: ((o)(o(o(o))))
%e A331963   334: (o((o)(o(o))))
%e A331963   501: ((o)((o)(o(o))))
%e A331963   606: (o(o)(o(o(o))))
%e A331963   794: (o(o(o)(o(o))))
%t A331963 msiQ[n_]:=n==1||n==2||!PrimeQ[n]&&SquareFreeQ[n]&&And@@msiQ/@PrimePi/@First/@FactorInteger[n];
%t A331963 Select[Range[1000],msiQ]
%Y A331963 A subset of A276625 (MG-numbers of identity trees).
%Y A331963 Not requiring an identity tree gives A331935.
%Y A331963 The locally disjoint version is A331937.
%Y A331963 These trees are counted by A331964.
%Y A331963 The semi-identity case is A331994.
%Y A331963 Matula-Goebel numbers of identity trees are A276625.
%Y A331963 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees are A331965.
%Y A331963 Cf. A004111, A007097, A050381, A061775, A196050, A291636, A300660, A306202, A320269, A331681, A331873, A331875, A331933, A331934, A331936.
%K A331963 nonn
%O A331963 1,2
%A A331963 _Gus Wiseman_, Feb 03 2020