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A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.

%I A318186 #6 Aug 22 2018 08:33:05
%S A318186 1,2,4,6,8,12,14,16,18,24,28,32,36,38,42,48,54,56,64,72,76,78,84,96,
%T A318186 98,106,108,112,114,126,128,144,152,156,162,168,192,196,212,216,222,
%U A318186 224,228,234,252,256,262,266,288,294,304,312,318,324,336,342,366,378
%N A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.
%C A318186 A number x is totally transitive if (1) whenever prime(y) divides x it follows that y is totally transitive and (2) if prime(y) divides x and prime(z) divides y then prime(z) also divides x.
%e A318186 The sequence of all totally transitive rooted trees together with their Matula-Goebel numbers begins:
%e A318186    1: o
%e A318186    2: (o)
%e A318186    4: (oo)
%e A318186    6: (o(o))
%e A318186    8: (ooo)
%e A318186   12: (oo(o))
%e A318186   14: (o(oo))
%e A318186   16: (oooo)
%e A318186   18: (o(o)(o))
%e A318186   24: (ooo(o))
%e A318186   28: (oo(oo))
%e A318186   32: (ooooo)
%e A318186   36: (oo(o)(o))
%e A318186   38: (o(ooo))
%e A318186   42: (o(o)(oo))
%e A318186   48: (oooo(o))
%e A318186   54: (o(o)(o)(o))
%e A318186   56: (ooo(oo))
%e A318186   64: (oooooo)
%e A318186   72: (ooo(o)(o))
%e A318186   76: (oo(ooo))
%e A318186   78: (o(o)(o(o)))
%e A318186   84: (oo(o)(oo))
%e A318186   96: (ooooo(o))
%e A318186   98: (o(oo)(oo))
%t A318186 subprimes[n_]:=If[n==1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
%t A318186 trmgQ[n_]:=Or[n==1,And[Divisible[n,Times@@subprimes[n]],And@@Cases[FactorInteger[n],{p_,_}:>trmgQ[PrimePi[p]]]]];
%t A318186 Select[Range[100],trmgQ]
%Y A318186 Cf. A000081, A001678, A004111, A007097, A061775, A276625, A279861, A290689, A290760, A290822, A291636, A318185, A318187.
%K A318186 nonn
%O A318186 1,2
%A A318186 _Gus Wiseman_, Aug 20 2018