In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients.
All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that Venkataraman (1975) found another family of perfect totient numbers: if p = 4 × 3k + 1 is prime, then 3p is 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.
This article incorporates material from Perfect Totient Number on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.