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Matula-Goebel numbers of rooted identity trees whose non-leaf terminal subtrees are all different.
5

%I #6 Mar 22 2019 00:33:28

%S 1,2,3,5,6,10,11,13,22,26,29,31,41,58,62,79,82,101,109,127,158,179,

%T 202,218,254,271,293,358,401,421,542,547,586,599,709,802,842,929,1063,

%U 1094,1198,1231,1361,1418,1609,1741,1858,1913,2126,2411,2462,2722,2749

%N Matula-Goebel numbers of rooted identity trees whose non-leaf terminal subtrees are all different.

%C A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. This sequence ranks rooted identity trees satisfying the additional condition that all non-leaf terminal subtrees are different.

%H Gus Wiseman, <a href="/A324968/a324968.png">The first 36 trees together with their Matula-Goebel numbers</a>.

%F Intersection of A324935 and A276625.

%e The sequence of trees together with the Matula-Goebel numbers begins:

%e 1: o

%e 2: (o)

%e 3: ((o))

%e 5: (((o)))

%e 6: (o(o))

%e 10: (o((o)))

%e 11: ((((o))))

%e 13: ((o(o)))

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

%e 26: (o(o(o)))

%e 29: ((o((o))))

%e 31: (((((o)))))

%e 41: (((o(o))))

%e 58: (o(o((o))))

%e 62: (o((((o)))))

%e 79: ((o(((o)))))

%e 82: (o((o(o))))

%e 101: ((o(o(o))))

%e 109: (((o((o)))))

%e 127: ((((((o))))))

%t mgtree[n_Integer]:=If[n==1,{},mgtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],And[And@@Cases[mgtree[#],q:{__}:>UnsameQ@@q,{0,Infinity}],UnsameQ@@Cases[mgtree[#],{__},{0,Infinity}]]&]

%Y Cf. A000081, A004111, A007097, A196050, A276625, A317713, A324850, A324923, A324935, A324936, A324969, A324970, A324978.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 21 2019