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

1,1

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

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.

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. A003434, A032358, A049108, A111974.

Sequence in context: A168614 A045014 A047591 * A239869 A289547 A329912

Adjacent sequences: A340762 A340763 A340764 * A340766 A340767 A340768

Keyword

nonn

Author

Franz Vrabec, Jan 20 2021

Status

approved