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