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, so 9 is a perfect totient number.
The first few perfect totient numbers are
In symbols, one writes
for the iterated totient function. Then if c is the integer such that
one has that n is a perfect totient number if

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
Venkataraman found another family of perfect totient numbers: if is prime, then 3p is a perfect totient number. The values of k leading to perfect totient numbers in this way are
More generally if p is a prime number greater than 3, and 3p is a perfect totient number, then p ≡ 1 . Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. 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.