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A324978 Matula-Goebel numbers of rooted trees that are not identity trees but whose non-leaf terminal subtrees are all different.

%I A324978 #6 Mar 22 2019 00:33:56
%S A324978 4,7,8,12,14,16,17,19,20,21,24,28,32,34,35,37,38,40,42,43,44,48,51,52,
%T A324978 53,56,57,59,64,67,68,70,71,73,74,76,77,80,84,85,86,88,89,91,95,96,
%U A324978 102,104,106,107,112,114,116,118,124,128,129,131,133,134,136,139
%N A324978 Matula-Goebel numbers of rooted trees that are not identity trees but whose non-leaf terminal subtrees are all different.
%C A324978 An unlabeled rooted tree is an identity tree if there are no repeated branches directly under the same root.
%H A324978 Gus Wiseman, <a href="/A324978/a324978.png">The first 36 trees together with their Matula-Goebel numbers</a>.
%F A324978 Complement of A276625 in A324935.
%e A324978 The sequence of trees together with the Matula-Goebel numbers begins:
%e A324978    4: (oo)
%e A324978    7: ((oo))
%e A324978    8: (ooo)
%e A324978   12: (oo(o))
%e A324978   14: (o(oo))
%e A324978   16: (oooo)
%e A324978   17: (((oo)))
%e A324978   19: ((ooo))
%e A324978   20: (oo((o)))
%e A324978   21: ((o)(oo))
%e A324978   24: (ooo(o))
%e A324978   28: (oo(oo))
%e A324978   32: (ooooo)
%e A324978   34: (o((oo)))
%e A324978   35: (((o))(oo))
%e A324978   37: ((oo(o)))
%e A324978   38: (o(ooo))
%e A324978   40: (ooo((o)))
%e A324978   42: (o(o)(oo))
%e A324978   43: ((o(oo)))
%t A324978 mgtree[n_]:=If[n==1,{},mgtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A324978 Select[Range[100],And[!And@@Cases[mgtree[#],q:{__}:>UnsameQ@@q,{0,Infinity}],UnsameQ@@Cases[mgtree[#],{__},{0,Infinity}]]&]
%Y A324978 Cf. A000081, A004111, A007097, A196050, A276625, A317713, A324850, A324923, A324935, A324936, A324968, A324969, A324970, A324971, A324979.
%K A324978 nonn
%O A324978 1,1
%A A324978 _Gus Wiseman_, Mar 21 2019