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Matula-Goebel numbers of rooted trees in which all outdegrees are either 0, 1, or 3.
2

%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