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.

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

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

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 (1975) found another family of perfect totient numbers: if p = 4 × 3^k + 1 is prime, then 3p is a perfect totient number. The values of k leading to perfect totient numbers in this way are