iterated totient function

The iterated totient function ϕk(n) is ak in the recurrence relation a0=n and ai=ϕ(ai-1) for i>0, where ϕ(x) is Euler’s totient function.

After enough iterations, the function eventually hits 2 followed by an infiniteMathworldPlanetmath trail of ones. Ianucci et al define the “class” c of n as the integer such that ϕc(n)=2.

When the iterated totient function is summed thus:


it can be observed that just as 2x is a quasiperfect number when it comes to adding up proper divisors, it is also “quasiperfect” when adding up the iterated totient function. Quite unlike regularPlanetmathPlanetmath perfect numbers, 3x (which are obviously odd) are “perfect” when adding up the iterated totient.


Title iterated totient function
Canonical name IteratedTotientFunction
Date of creation 2013-03-22 16:33:09
Last modified on 2013-03-22 16:33:09
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 5
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A25
Related topic PerfectTotientNumber