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A297571 Matula-Goebel numbers of fully unbalanced rooted trees.

%I A297571 #8 Jan 07 2018 23:35:34
%S A297571 1,2,3,5,6,10,11,13,15,22,26,29,30,31,33,39,41,47,55,58,62,65,66,78,
%T A297571 79,82,87,93,94,101,109,110,113,123,127,130,137,141,145,155,158,165,
%U A297571 167,174,179,186,195,202,205,211,218,226,235,237,246,254,257,271,274
%N A297571 Matula-Goebel numbers of fully unbalanced rooted trees.
%C A297571 An unlabeled rooted tree is fully unbalanced if either (1) it is a single node, or (2a) every branch has a different number of nodes and (2b) every branch is fully unbalanced also. The number of fully unbalanced trees with n nodes is A032305(n).
%C A297571 The first finitary number (A276625) not in this sequence is 143.
%e A297571 Sequence of fully unbalanced trees begins:
%e A297571    1 o
%e A297571    2 (o)
%e A297571    3 ((o))
%e A297571    5 (((o)))
%e A297571    6 (o(o))
%e A297571   10 (o((o)))
%e A297571   11 ((((o))))
%e A297571   13 ((o(o)))
%e A297571   15 ((o)((o)))
%e A297571   22 (o(((o))))
%e A297571   26 (o(o(o)))
%e A297571   29 ((o((o))))
%e A297571   30 (o(o)((o)))
%e A297571   31 (((((o)))))
%e A297571   33 ((o)(((o))))
%e A297571   39 ((o)(o(o)))
%e A297571   41 (((o(o))))
%e A297571   47 (((o)((o))))
%t A297571 nn=2000;
%t A297571 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A297571 MGweight[n_]:=If[n===1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>k*MGweight[PrimePi[p]]]]];
%t A297571 imbalQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[UnsameQ@@MGweight/@m,And@@imbalQ/@m]]];
%t A297571 Select[Range[nn],imbalQ]
%Y A297571 Cf. A000081, A003238, A004111, A007097, A032305, A061775, A214577, A273873, A276625, A277098, A290689, A290760, A291441, A291442, A291443.
%K A297571 nonn
%O A297571 1,2
%A A297571 _Gus Wiseman_, Dec 31 2017