A340765 Numbers k such that iterations of phi(k), phi(phi(k)), ... end in ... 6, 2, 1.

6, 7, 9, 14, 18, 19, 27, 38, 54, 81, 162, 163, 243, 326, 486, 487, 729, 974, 1458, 1459, 2187, 2918, 4374, 6561, 13122, 19683, 39366, 39367, 59049, 78734, 118098, 177147, 354294, 531441, 1062882, 1594323, 3188646, 4782969, 9565938, 14348907, 28697814, 43046721, 86093442, 86093443, 129140163, 172186886

Offset: 1

Author

Franz Vrabec, Jan 20 2021

Keywords

• nonn

Comments

Infinite set (see reference).

Contains 3^k for k >= 2 and 2*3^k for k >= 1, and all members of A111974 except 3. - Robert Israel, Dec 23 2021

Examples

			19 is in the list because phi(phi(19)) = phi(18) = 6.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000
• Eliot T. Jacobson and Alan E. Parks, Infinite branches of the phi-tree, Amer. Math. Monthly, Vol. 93, No. 7 (August-September 1986), pp. 552-554.
• Keith Matthews, Solving phi(x)=n, where phi(x) is Euler's totient function.

Crossrefs

Cf. A000010, A340762 (complement relative to {n>=4}).

Cf. A003434, A032358, A049108, A111974.

Programs

Maple

R:= {6}: Agenda:= {6}: count:= 1:
while count - nops(Agenda) < 99 do
  v:= min(Agenda);
  W:= convert(numtheory:-invphi(v),set);
  count:= count + nops(W);
  Agenda:= Agenda minus {v} union W;
  R:= R union W;
od:
sort(select(`<=`, convert(R,list),min(Agenda))); # Robert Israel, Dec 23 2021

Mathematica

Select[Range[4, 10000], FixedPointList[EulerPhi, #][[-4]] == 6 &] (* Amiram Eldar, Jan 27 2021 *)

PARI

isok(k) = if (k>=6, while((k!=6) && (k!=4), k=eulerphi(k))); k == 6; \\ Michel Marcus, Feb 01 2021