A049108 a(n) is the number of iterations of Euler phi function needed to reach 1 starting at n (n is counted).

1, 2, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 5, 5, 6, 4, 5, 5, 5, 5, 6, 5, 6, 5, 5, 5, 6, 5, 6, 6, 6, 6, 6, 5, 6, 5, 6, 6, 7, 5, 6, 6, 6, 6, 7, 6, 6, 6, 7, 6, 7, 5, 7, 6, 6, 6, 7, 6, 7, 6, 6, 7, 7, 6, 7, 7, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 7, 6, 7, 8, 6, 8, 6, 7, 7, 8, 6, 7, 7, 7, 7, 7, 7, 8, 6, 7, 7, 8, 7, 8, 7, 7

Offset: 1

Author

Labos Elemer

Keywords

• nonn
• nice
• easy

Examples

			If n=164 the trajectory is {164,80,32,16,8,4,2,1}. Its length is 8, thus a(164)=8.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A007755. Equals A003434 + 1. Row lengths of A375478.

Programs

Maple

A049108 := proc(n)
    local a, e;
    e := n ;
    a :=0 ;
    while e > 1 do
        a := a+1 ;
        e := numtheory[phi](e) ;
    end do:
    1+a;
end proc:
seq(A049108(n),n=1..60) ; # R. J. Mathar, Sep 08 2021

Mathematica

f[n_] := Length[NestWhileList[ EulerPhi, n, # != 1 &]]; Array[f, 105] (* Robert G. Wilson v, Feb 07 2012 *)

PARI

a(n)=my(t=1);while(n>1,t++;n=eulerphi(n));t \\ Charles R Greathouse IV, Feb 07 2012

Formula

By the definition of a(n) we have for n >= 2 the recursion a(n) = a(Phi(n)) + 1. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001

log_3 n << a(n) << log_2 n. - Charles R Greathouse IV, Feb 07 2012