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, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.

Examples
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, ....

Notation
In symbols, one writes
 * $$\varphi^i(n) = \begin{cases}

\varphi(n), &\text{ if } i = 1 \\ \varphi(\varphi^{i-1}(n)), &\text{ if } i \geq 2 \end{cases}$$ for the iterated totient function. Then if c is the integer such that
 * $$\displaystyle\varphi^c(n)=2,$$

one has that n is a perfect totient number if
 * $$n = \sum_{i = 1}^{c + 1} \varphi^i(n).$$

Multiples and powers of three
It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that
 * $$\displaystyle\varphi(3^k) = \varphi(2\times 3^k) = 2 \times 3^{k-1}.$$

Venkataraman (1975) found another family of perfect totient numbers: if $p = 4&thinsp;×&thinsp;3^{k} +&thinsp;1$ is prime, then 3p is a perfect totient number. The values of k leading to perfect totient numbers in this way are
 * 0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ....

More generally if p is a prime number greater than 3, and 3p is a perfect totient number, then p ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9p is a perfect totient number then p is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers of the form 3kp where p is prime and k > 3.