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A324935 Matula-Goebel numbers of rooted trees whose non-leaf terminal subtrees are all different.

%I A324935 #5 Mar 22 2019 00:33:01
%S A324935 1,2,3,4,5,6,7,8,10,11,12,13,14,16,17,19,20,21,22,24,26,28,29,31,32,
%T A324935 34,35,37,38,40,41,42,43,44,48,51,52,53,56,57,58,59,62,64,67,68,70,71,
%U A324935 73,74,76,77,79,80,82,84,85,86,88,89,91,95,96,101,102,104
%N A324935 Matula-Goebel numbers of rooted trees whose non-leaf terminal subtrees are all different.
%C A324935 Every positive integer has a unique factorization into factors q(i) = prime(i)/i, i > 0. This sequence consists of all numbers where this factorization has all distinct factors, except possibly for any multiplicity of q(1). For example, 22 = q(1)^2 q(2) q(3) q(5) is in the sequence, while 50 = q(1)^3 q(2)^2 q(3)^2 is not.
%C A324935 The enumeration of these trees by number of vertices is A324936.
%H A324935 Gus Wiseman, <a href="/A324935/a324935.png">The first 36 rooted trees whose non-leaf terminal subtrees are all different, together with their Matula-Goebel numbers</a>.
%e A324935 The sequence of trees together with their Matula-Goebel numbers begins:
%e A324935    1: o
%e A324935    2: (o)
%e A324935    3: ((o))
%e A324935    4: (oo)
%e A324935    5: (((o)))
%e A324935    6: (o(o))
%e A324935    7: ((oo))
%e A324935    8: (ooo)
%e A324935   10: (o((o)))
%e A324935   11: ((((o))))
%e A324935   12: (oo(o))
%e A324935   13: ((o(o)))
%e A324935   14: (o(oo))
%e A324935   16: (oooo)
%e A324935   17: (((oo)))
%e A324935   19: ((ooo))
%e A324935   20: (oo((o)))
%e A324935   21: ((o)(oo))
%e A324935   22: (o(((o))))
%e A324935   24: (ooo(o))
%e A324935   26: (o(o(o)))
%e A324935   28: (oo(oo))
%e A324935   29: ((o((o))))
%e A324935   31: (((((o)))))
%t A324935 difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
%t A324935 Select[Range[100],UnsameQ@@DeleteCases[difac[#],1]&]
%Y A324935 Cf. A000081, A004111, A007097, A061775, A196050, A276625, A290822, A317713.
%Y A324935 Cf. A324850, A324922, A324923, A324924, A324931, A324934, A324936.
%K A324935 nonn
%O A324935 1,2
%A A324935 _Gus Wiseman_, Mar 21 2019