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%I #6 Jan 22 2018 03:07:40
%S 1,2,3,5,7,11,13,14,17,19,21,23,26,29,31,34,35,37,38,39,41,43,46,47,
%T 51,53,57,58,59,61,65,67,69,71,73,74,77,79,82,83,85,86,87,89,94,95,97,
%U 101,103,106,107,109,111,113,115,118,122,123,127,129,131,133
%N Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of leaves.
%e Sequence of trees begins:
%e 1 o
%e 2 (o)
%e 3 ((o))
%e 5 (((o)))
%e 7 ((oo))
%e 11 ((((o))))
%e 13 ((o(o)))
%e 14 (o(oo))
%e 17 (((oo)))
%e 19 ((ooo))
%e 21 ((o)(oo))
%e 23 (((o)(o)))
%e 26 (o(o(o)))
%e 29 ((o((o))))
%e 31 (((((o)))))
%e 34 (o((oo)))
%e 35 (((o))(oo))
%e 37 ((oo(o)))
%e 38 (o(ooo))
%e 39 ((o)(o(o)))
%e 41 (((o(o))))
%e 43 ((o(oo)))
%e 46 (o((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 leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
%t Select[Range[nn],UnsameQ@@leafcount/@primeMS[#]&]
%Y Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290689, A290760, A291442, A298534, A298535.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 20 2018
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