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A290822 Transitive numbers: Matula-Goebel numbers of transitive rooted trees.

%I A290822 #26 Dec 18 2023 09:38:40
%S A290822 1,2,4,6,8,12,14,16,18,24,28,30,32,36,38,42,48,54,56,60,64,72,76,78,
%T A290822 84,90,96,98,106,108,112,114,120,126,128,138,144,150,152,156,162,168,
%U A290822 180,192,196,210,212,216,222,224,228,234,238,240,252,256,262,266,270
%N A290822 Transitive numbers: Matula-Goebel numbers of transitive rooted trees.
%C A290822 A number x is transitive if whenever prime(y) divides x and prime(z) divides y, we have prime(z) divides x.
%H A290822 Robert P. P. McKone, <a href="/A290822/b290822.txt">Table of n, a(n) for n = 1..9999</a>
%e A290822 The sequence of transitive rooted trees begins:
%e A290822 1  o
%e A290822 2  (o)
%e A290822 4  (oo)
%e A290822 6  (o(o))
%e A290822 8  (ooo)
%e A290822 12 (oo(o))
%e A290822 14 (o(oo))
%e A290822 16 (oooo)
%e A290822 18 (o(o)(o))
%e A290822 24 (ooo(o))
%e A290822 28 (oo(oo))
%e A290822 30 (o(o)((o)))
%e A290822 32 (ooooo)
%e A290822 36 (oo(o)(o))
%e A290822 38 (o(ooo))
%e A290822 42 (o(o)(oo))
%e A290822 48 (oooo(o))
%e A290822 54 (o(o)(o)(o))
%e A290822 56 (ooo(oo))
%e A290822 60 (oo(o)((o)))
%e A290822 64 (oooooo)
%e A290822 72 (ooo(o)(o))
%e A290822 76 (oo(ooo))
%e A290822 78 (o(o)(o(o)))
%e A290822 84 (oo(o)(oo))
%e A290822 90 (o(o)(o)((o)))
%e A290822 96 (ooooo(o))
%e A290822 98 (o(oo)(oo))
%t A290822 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A290822 subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
%t A290822 Select[Range[270],Divisible[#,Times@@subprimes[#]]&]
%Y A290822 Cf. A007097, A276625, A279861, A290689, A290760.
%K A290822 nonn
%O A290822 1,2
%A A290822 _Gus Wiseman_, Oct 19 2017