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A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees. 7
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 (list; graph; refs; listen; history; text; internal format)
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 A275692

Adjacent sequences:  A318183 A318184 A318185 * A318187 A318188 A318189

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 20 2018

STATUS

approved

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Last modified June 25 10:29 EDT 2019. Contains 324351 sequences. (Running on oeis4.)