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%I #15 Jun 26 2021 06:45:22
%S 1,4,14,86,301,886,3101,3986,13766,13951,19049,48181,57026,75266,
%T 85699,199591,263431,295969,298154,302426,426058,882899
%N Matula-Goebel numbers of unlabeled binary rooted semi-identity trees.
%C Definition: A positive integer belongs to the sequence iff it is 1, 4, or a squarefree semiprime whose prime indices both already belong to the sequence. 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 In a semi-identity tree, only the non-leaf branches of any given vertex are distinct. Alternatively, 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.
%C The Matula-Goebel number of an unlabeled 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.
%H Gus Wiseman, <a href="/A339193/a339193.png">The sequence of all unlabeled binary rooted semi-identity trees by Matula-Goebel number.</a>
%e The sequence of terms together with the corresponding unlabeled rooted trees begins:
%e 1: o
%e 4: (oo)
%e 14: (o(oo))
%e 86: (o(o(oo)))
%e 301: ((oo)(o(oo)))
%e 886: (o(o(o(oo))))
%e 3101: ((oo)(o(o(oo))))
%e 3986: (o((oo)(o(oo))))
%e 13766: (o(o(o(o(oo)))))
%e 13951: ((oo)((oo)(o(oo))))
%e 19049: ((o(oo))(o(o(oo))))
%e 48181: ((oo)(o(o(o(oo)))))
%e 57026: (o((oo)(o(o(oo)))))
%e 75266: (o(o((oo)(o(oo)))))
%e 85699: ((o(oo))((oo)(o(oo))))
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t mgbiQ[n_]:=Or[n==1,n==4,SquareFreeQ[n]&&PrimeOmega[n]==2&&And@@mgbiQ/@primeMS[n]];
%t Select[Range[1000],mgbiQ]
%Y Counting these trees by number of nodes gives A063895.
%Y A000081 counts unlabeled rooted trees with n nodes.
%Y A111299 ranks binary trees, counted by A001190.
%Y A276625 ranks identity trees, counted by A004111.
%Y A306202 ranks semi-identity trees, counted by A306200.
%Y A306203 ranks balanced semi-identity trees, counted by A306201.
%Y A331965 ranks lone-child avoiding semi-identity trees, counted by A331966.
%Y Cf. A007097, A061775, A196050, A291636, A331963, A331964.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 14 2021
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