%I #5 Jan 19 2018 02:07:26
%S 1,2,3,5,6,10,11,15,22,26,30,31,33,39,55,58,62,65,66,78,87,93,94,110,
%T 127,130,141,143,145,155,158,165,174,186,195,202,235,237,254,274,282,
%U 286,290,303,310,319,330,334,341,377,381,390,395,403,411,429,435,465
%N Matula-Goebel numbers of rooted identity trees with thinning limbs.
%C An unlabeled rooted tree has thinning limbs if its outdegrees are weakly decreasing from root to leaves.
%F Intersection of A276625 and A298303.
%e Sequence of 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 15 ((o)((o)))
%e 22 (o(((o))))
%e 26 (o(o(o)))
%e 30 (o(o)((o)))
%e 31 (((((o)))))
%e 33 ((o)(((o))))
%e 39 ((o)(o(o)))
%e 55 (((o))(((o))))
%e 58 (o(o((o))))
%e 62 (o((((o)))))
%e 65 (((o))(o(o)))
%e 66 (o(o)(((o))))
%e 78 (o(o)(o(o)))
%e 87 ((o)(o((o))))
%e 93 ((o)((((o)))))
%e 94 (o((o)((o))))
%t MGtree[n_]:=If[n===1,{},MGtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t idthinQ[t_]:=And@@Cases[t,b_List:>UnsameQ@@b&&Length[b]>=Max@@Length/@b,{0,Infinity}];
%t Select[Range[500],idthinQ[MGtree[#]]&]
%Y Cf. A000081, A004111, A007097, A061775, A111299, A124343, A124346, A214577, A276625, A277098, A290689, A290760, A298304, A298305.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 17 2018