Perfect totient number

In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.

For example, there are six positive integers less than 9 and relatively prime to it, so the totient of 9 is 6; there are two numbers less than 6 and relatively prime to it, so the totient of 6 is 2; and there is one number less than 2 and relatively prime to it, so the totient of 2 is 1; and 9 = 6 + 2 + 1, so 9 is a perfect totient number.

The first few perfect totient numbers are

3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, ... (sequence A082897 in the OEIS).

In symbols, one writes

${\displaystyle \varphi ^{i}(n)=\left\{{\begin{matrix}\varphi (n)&{\mbox{ if }}i=1\\\varphi (\varphi ^{i-1}(n))&{\mbox{ otherwise}}\end{matrix}}\right.}$

for the iterated totient function. Then if c is the integer such that

${\displaystyle \displaystyle \varphi ^{c}(n)=2,}$

one has that n is a perfect totient number if

${\displaystyle n=\sum _{i=1}^{c+1}\varphi ^{i}(n).}$