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, 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, ... .
In symbols, one writes
\varphii(n)=\begin{cases} \varphi(n),&ifi=1\\ \varphi(\varphii-1(n)),&ifi\geq2 \end{cases}
\displaystyle\varphic(n)=2,
n=
| c+1 | |
| \sum | |
| i=1 |
\varphii(n).
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(3k)=\varphi(2 x 3k)=2 x 3k-1.
Venkataraman (1975) 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
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.