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 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, ... (sequence A082897 in the OEIS).
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 (1975) found another family of perfect totient numbers: if p = 4 × 3k + 1 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 (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.
- Pérez-Cacho Villaverde, Laureano (1939). "Sobre la suma de indicadores de ordenes sucesivos". Revista Matematica Hispano-Americana. 5 (3): 45–50.
- Guy, Richard K. (2004). Unsolved Problems in Number Theory. New York: Springer-Verlag. p. §B41. ISBN 0-387-20860-7.
- Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L. (2003). "On perfect totient numbers" (PDF). Journal of Integer Sequences. 6 (4): 03.4.5. MR 2051959.
- Luca, Florian (2006). "On the distribution of perfect totients" (PDF). Journal of Integer Sequences. 9 (4): 06.4.4. MR 2247943. Archived from the original (PDF) on 2017-08-11. Retrieved 2007-02-07.
- Mohan, A. L.; Suryanarayana, D. (1982). "Perfect totient numbers". Number theory (Mysore, 1981). Lecture Notes in Mathematics, vol. 938, Springer-Verlag. pp. 101–105. MR 0665442.