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

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 32, 36, 38, 42, 48, 54, 56, 64, 72, 76, 78, 84, 96, 98, 106, 108, 112, 114, 126, 128, 144, 152, 156, 162, 168, 192, 196, 212, 216, 222, 224, 228, 234, 252, 256, 262, 266, 288, 294, 304, 312, 318, 324, 336, 342, 366, 378

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..58.

Example

The sequence of all totally transitive rooted trees together with their Matula-Goebel numbers 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))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  76: (oo(ooo))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))
  98: (o(oo)(oo))

Mathematica

subprimes[n_]:=If[n==1, {}, Union@@Cases[FactorInteger[n], {p_, _}:>FactorInteger[PrimePi[p]][[All, 1]]]];
trmgQ[n_]:=Or[n==1, And[Divisible[n, Times@@subprimes[n]], And@@Cases[FactorInteger[n], {p_, _}:>trmgQ[PrimePi[p]]]]];
Select[Range[100], trmgQ]

Crossrefs

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

Sequence in context: A088879 A316470 A290822 * A139363 A091065 A328596

Adjacent sequences: A318183 A318184 A318185 * A318187 A318188 A318189

Keyword

nonn

Author

Gus Wiseman, Aug 20 2018

Status

approved