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%I #5 Jan 22 2018 03:08:06
%S 1,2,3,5,6,7,10,11,13,14,15,17,19,21,22,23,26,29,30,31,33,34,37,38,39,
%T 41,42,43,46,47,51,53,55,57,58,59,61,62,65,66,67,69,71,73,74,77,78,79,
%U 82,83,85,86,87,89,91,93,94,95,97,101,102,103,106,107,109
%N Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of nodes.
%e Sequence of trees begins:
%e 1 o
%e 2 (o)
%e 3 ((o))
%e 5 (((o)))
%e 6 (o(o))
%e 7 ((oo))
%e 10 (o((o)))
%e 11 ((((o))))
%e 13 ((o(o)))
%e 14 (o(oo))
%e 15 ((o)((o)))
%e 17 (((oo)))
%e 19 ((ooo))
%e 21 ((o)(oo))
%e 22 (o(((o))))
%e 23 (((o)(o)))
%e 26 (o(o(o)))
%e 29 ((o((o))))
%e 30 (o(o)((o)))
%t nn=500;
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t MGweight[n_]:=If[n===1,1,1+Total[MGweight/@primeMS[n]]];
%t Select[Range[nn],UnsameQ@@MGweight/@primeMS[#]&]
%Y Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A291442, A297571, A298479, A298534, A298536, A298538, A298539.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 21 2018
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