A073408 Let cophi_m(x) denotes the cototient function applied m times to x (cophi(x)=x-phi(x)). Sequence gives the minimum number of iterations m such that cophi_m(n) divides n.
1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 3, 2, 1, 1, 4, 1, 3, 2, 4, 1, 2, 1, 4, 1, 3, 1, 5, 1, 1, 2, 5, 2, 4, 1, 5, 3, 3, 1, 6, 1, 4, 2, 5, 1, 2, 1, 6, 2, 4, 1, 6, 3, 3, 3, 6, 1, 5, 1, 5, 2, 1, 2, 6, 1, 5, 3, 6, 1, 4, 1, 6, 3, 5, 2, 7, 1, 3, 1, 7, 1, 6, 4, 6, 2, 4, 1, 7, 2, 5, 3, 6, 2, 2, 1, 6, 4, 6, 1, 7, 1, 4, 2, 7
Offset: 2
Examples
cophi(10) -> 6, cophi(6) -> 4, cophi(4) -> 2 and 2 divides 10. Hence 3 iterations are needed and a(10) = 3.
Links
- Antti Karttunen, Table of n, a(n) for n = 2..16385
Programs
-
Mathematica
Table[Length@ NestWhileList[# - EulerPhi@ # &, n, Or[# == n, ! Divisible[n, #]] &, 1, 12] - 1, {n, 2, 106}] (* Michael De Vlieger, Dec 22 2017 *) -
PARI
a(n)=if(n<0,0,c=1; s=n; while(n%(s-eulerphi(s))>0,s=s-eulerphi(s); c++); c)
Formula
It seems that sum(k=1, n, a(k)) is asymptotic to C*n*log(n) with C < 1.