A289125 Numbers n such that phi(n)/phi(phi(n)) > phi(m)/phi(phi(m)) for all m < n.

1, 3, 7, 31, 211, 2311, 43891, 60653, 870871, 1023053, 13123111, 19417793, 300690391, 446235509, 6915878971, 12939711677, 200560490131

Offset: 1

Author

Amiram Eldar, Jun 25 2017

Keywords

• nonn
• more

Comments

Erdős et al. proved that phi(n)/phi(phi(n)) is unbounded, thus this sequence is infinite.

A018239(k) = A002110(A014545(k)) + 1 is a term for k > 1. Are there terms m with omega(m) > 2? Is omega(phi(a(n + 1))) >= omega(phi(a(n)))? - David A. Corneth, Jun 28 2017

Links

• Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
• Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]

Crossrefs

Cf. A000010, A001221, A002110, A010554, A014545, A018239.

Programs

Mathematica

a = {}; k=1; rmax = 0; While[Length[a]<10,s = EulerPhi[ k]; s2 = EulerPhi[ s]; r = s/s2;  If[r > rmax, AppendTo[a, k]; rmax = r]; k++]; a
DeleteDuplicates[Table[{n,EulerPhi[n]/EulerPhi[EulerPhi[n]]},{n,11*10^5}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first ten terms of the sequence. *) (* Harvey P. Dale, Aug 17 2024 *)

PARI

r=0; forfactored(n=1,10^10, t=eulerphi(n); t/=eulerphi(t); if(t>r, r=t; print1(n[1]", "))) \\ Charles R Greathouse IV, Jun 25 2017

Extensions

a(15)-a(17) from Giovanni Resta, Jul 01 2017