A318186 - OEIS

%I #6 Aug 22 2018 08:33:05

%S 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 98,106,108,112,114,126,128,144,152,156,162,168,192,196,212,216,222,

%U 224,228,234,252,256,262,266,288,294,304,312,318,324,336,342,366,378

%N Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.

%C 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 The sequence of all totally transitive 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   14: (o(oo))

%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   38: (o(ooo))

%e   42: (o(o)(oo))

%e   48: (oooo(o))

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

%e   56: (ooo(oo))

%e   64: (oooooo)

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

%e   76: (oo(ooo))

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

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

%e   96: (ooooo(o))

%e   98: (o(oo)(oo))

%t subprimes[n_]:=If[n==1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];

%t trmgQ[n_]:=Or[n==1,And[Divisible[n,Times@@subprimes[n]],And@@Cases[FactorInteger[n],{p_,_}:>trmgQ[PrimePi[p]]]]];

%t Select[Range[100],trmgQ]

%Y Cf. A000081, A001678, A004111, A007097, A061775, A276625, A279861, A290689, A290760, A290822, A291636, A318185, A318187.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 20 2018