%I #7 Mar 22 2019 00:33:41
%S 15,30,33,39,47,55,65,66,78,87,93,94,110,113,123,130,137,141,143,145,
%T 155,165,167,174,186,195,205,211,226,235,237,246,257,274,282,286,290,
%U 303,310,313,317,319,327,330,334,339,341,377,381,390,395,397,403,410
%N Matula-Goebel numbers of rooted identity trees where not all terminal subtrees are different.
%C A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root.
%H Gus Wiseman, <a href="/A324970/a324970.png">The first 36 trees together with their Matula-Goebel numbers</a>.
%F Complement of A324935 in A276625.
%e The sequence of trees together with the Matula-Goebel numbers begins:
%e 15: ((o)((o)))
%e 30: (o(o)((o)))
%e 33: ((o)(((o))))
%e 39: ((o)(o(o)))
%e 47: (((o)((o))))
%e 55: (((o))(((o))))
%e 65: (((o))(o(o)))
%e 66: (o(o)(((o))))
%e 78: (o(o)(o(o)))
%e 87: ((o)(o((o))))
%e 93: ((o)((((o)))))
%e 94: (o((o)((o))))
%e 110: (o((o))(((o))))
%e 113: ((o(o)((o))))
%e 123: ((o)((o(o))))
%e 130: (o((o))(o(o)))
%e 137: (((o)(((o)))))
%e 141: ((o)((o)((o))))
%e 143: ((((o)))(o(o)))
%e 145: (((o))(o((o))))
%e 155: (((o))((((o)))))
%e 165: ((o)((o))(((o))))
%e 167: (((o)(o(o))))
%e 174: (o(o)(o((o))))
%e 186: (o(o)((((o)))))
%e 195: ((o)((o))(o(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, A324968, A324971, A324978.
%K nonn
%O 1,1
%A _Gus Wiseman_, Mar 21 2019