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Matula-Goebel numbers of fully unbalanced rooted trees.
7

%I #8 Jan 07 2018 23:35:34

%S 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 79,82,87,93,94,101,109,110,113,123,127,130,137,141,145,155,158,165,

%U 167,174,179,186,195,202,205,211,218,226,235,237,246,254,257,271,274

%N Matula-Goebel numbers of fully unbalanced rooted trees.

%C 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 The first finitary number (A276625) not in this sequence is 143.

%e Sequence of fully unbalanced trees 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 15 ((o)((o)))

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

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

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

%e 30 (o(o)((o)))

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

%e 33 ((o)(((o))))

%e 39 ((o)(o(o)))

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

%e 47 (((o)((o))))

%t nn=2000;

%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[Cases[FactorInteger[n],{p_,k_}:>k*MGweight[PrimePi[p]]]]];

%t imbalQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[UnsameQ@@MGweight/@m,And@@imbalQ/@m]]];

%t Select[Range[nn],imbalQ]

%Y Cf. A000081, A003238, A004111, A007097, A032305, A061775, A214577, A273873, A276625, A277098, A290689, A290760, A291441, A291442, A291443.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 31 2017