%I #7 Jan 22 2018 03:07:29
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,15,16,17,18,19,20,22,23,24,25,27,29,30,
%T 31,32,33,36,37,40,41,43,44,45,47,48,49,50,53,54,55,59,60,61,62,64,66,
%U 67,71,72,73,75,79,80,81,83,88,89,90,91,93,96,97,99,100
%N Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of leaves.
%e Sequence of trees begins:
%e 1 o
%e 2 (o)
%e 3 ((o))
%e 4 (oo)
%e 5 (((o)))
%e 6 (o(o))
%e 7 ((oo))
%e 8 (ooo)
%e 9 ((o)(o))
%e 10 (o((o)))
%e 11 ((((o))))
%e 12 (oo(o))
%e 13 ((o(o)))
%e 15 ((o)((o)))
%e 16 (oooo)
%e 17 (((oo)))
%e 18 (o(o)(o))
%e 19 ((ooo))
%e 20 (oo((o)))
%e 22 (o(((o))))
%e 23 (((o)(o)))
%e 24 (ooo(o))
%e 25 (((o))((o)))
%e 27 ((o)(o)(o))
%e 29 ((o((o))))
%e 30 (o(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],SameQ@@leafcount/@primeMS[#]&]
%Y Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A290822, A291442, A298533, A298536.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 20 2018