%I #6 Jan 17 2018 04:28:29
%S 1,2,3,5,8,11,12,18,19,20,27,30,31,37,44,45,50,61,66,67,71,75,76,99,
%T 103,110,113,114,124,125,127,148,157,165,171,186,190,193,197,222,229,
%U 242,244,268,275,279,283,284,285,310,317,331,333,353,363,366,370,379
%N Matula-Goebel numbers of rooted trees in which all outdegrees are either 0, 1, or 3.
%e Sequence of rooted trees begins:
%e 1 o
%e 2 (o)
%e 3 ((o))
%e 5 (((o)))
%e 8 (ooo)
%e 11 ((((o))))
%e 12 (oo(o))
%e 18 (o(o)(o))
%e 19 ((ooo))
%e 20 (oo((o)))
%e 27 ((o)(o)(o))
%e 30 (o(o)((o)))
%e 31 (((((o)))))
%e 37 ((oo(o)))
%e 44 (oo(((o))))
%e 45 ((o)(o)((o)))
%e 50 (o((o))((o)))
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t stQ[n_]:=Or[n===1,With[{m=primeMS[n]},MemberQ[{1,3},Length[m]]&&And@@stQ/@m]];
%t Select[Range[10000],stQ]
%Y Cf. A000081, A000598, A014591, A026424, A032305, A061775, A111299, A276625, A292050, A295461, A297571, A298126, A298118, A298120, A298204, A298207.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jan 14 2018