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Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.
2

%I #5 Mar 19 2019 07:14:52

%S 1,2,4,6,8,12,16,18,24,28,32,36,48,54,56,64,72,78,84,96,108,112,128,

%T 144,152,156,162,168,192,196,216,224,234,252,256,288,304,312,324,336,

%U 384,392,432,444,448,456,468,486,504,512,576,588,608,624,648,672,702

%N Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

%e The sequence of rooted trees together with their Matula-Goebel numbers begins:

%e 1: o

%e 2: (o)

%e 4: (oo)

%e 6: (o(o))

%e 8: (ooo)

%e 12: (oo(o))

%e 16: (oooo)

%e 18: (o(o)(o))

%e 24: (ooo(o))

%e 28: (oo(oo))

%e 32: (ooooo)

%e 36: (oo(o)(o))

%e 48: (oooo(o))

%e 54: (o(o)(o)(o))

%e 56: (ooo(oo))

%e 64: (oooooo)

%e 72: (ooo(o)(o))

%e 78: (o(o)(o(o)))

%e 84: (oo(o)(oo))

%e 96: (ooooo(o))

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t qaQ[n_]:=And[And@@Table[Divisible[n,x],{x,primeMS[n]}],And@@qaQ/@primeMS[n]];

%t Select[Range[1000],qaQ]

%Y A subsequence of A120383.

%Y Cf. A000081, A007097, A112798, A279861, A290689, A290760, A290822, A318186.

%Y Cf. A324704, A324736, A324748, A324753, A324843, A324847, A324848, A324854.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 18 2019