proof that the sum of the iterated totient function is always odd


Given a positive integer n, it is always the case that

2i=1c+1ϕi(n),

where ϕi(x) is the iterated totient function and c is the integer such that ϕc(n)=2.

Accepting as proven that n>ϕ(n) and 2|ϕ(n) for n>2, it is clear that summing up the iterates of the totient function up to c is summing up a series of even numbersMathworldPlanetmath in descending order and that this sum is therefore itself even. Then, when we add the c+1 iterate, the sum turns odd.

As a bonus, this proves that no even number can be a perfect totient number.

Title proof that the sum of the iterated totient function is always odd
Canonical name ProofThatTheSumOfTheIteratedTotientFunctionIsAlwaysOdd
Date of creation 2013-03-22 16:34:26
Last modified on 2013-03-22 16:34:26
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Proof
Classification msc 11A25