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

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 30, 32, 36, 38, 42, 48, 54, 56, 60, 64, 72, 76, 78, 84, 90, 96, 98, 106, 108, 112, 114, 120, 126, 128, 138, 144, 150, 152, 156, 162, 168, 180, 192, 196, 210, 212, 216, 222, 224, 228, 234, 238, 240, 252, 256, 262, 266, 270

Offset: 1

Author

Gus Wiseman, Oct 19 2017

Keywords

• nonn

Comments

A number x is transitive if whenever prime(y) divides x and prime(z) divides y, we have prime(z) divides x.

Examples

			The sequence of transitive rooted trees begins:
1  o
2  (o)
4  (oo)
6  (o(o))
8  (ooo)
12 (oo(o))
14 (o(oo))
16 (oooo)
18 (o(o)(o))
24 (ooo(o))
28 (oo(oo))
30 (o(o)((o)))
32 (ooooo)
36 (oo(o)(o))
38 (o(ooo))
42 (o(o)(oo))
48 (oooo(o))
54 (o(o)(o)(o))
56 (ooo(oo))
60 (oo(o)((o)))
64 (oooooo)
72 (ooo(o)(o))
76 (oo(ooo))
78 (o(o)(o(o)))
84 (oo(o)(oo))
90 (o(o)(o)((o)))
96 (ooooo(o))
98 (o(oo)(oo))
		

Links

• Robert P. P. McKone, Table of n, a(n) for n = 1..9999

Crossrefs

Cf. A007097, A276625, A279861, A290689, A290760.

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
Select[Range[270],Divisible[#,Times@@subprimes[#]]&]