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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. Or to put it algebraically, if
## Multiples and powers of three

## References

- $n\; =\; sum\_\{i\; =\; 1\}^\{c\; +\; 1\}\; varphi^i(n),$

- $varphi^i(n)=left\{begin\{matrix\}varphi(n)\&mbox\{\; if\; \}\; i=1\; varphi(varphi^\{i-1\}(n))\&mbox\{\; otherwise\}end\{matrix\}right.$

- $displaystylevarphi^c(n)=2,$

The first few perfect totient numbers are

For example, start with 327. φ(327) = 216, φ(216) = 72, φ(72) = 24, φ(24) = 8, φ(8) = 4, φ(4) = 2, φ(2) = 1, and 216 + 72 + 24 + 8 + 4 + 2 + 1 = 327.

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

- $displaystylevarphi(3^k)\; =\; varphi(2times\; 3^k)\; =\; 2times\; 3^\{k-1\}.$

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

- 0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... .

More generally if p is a prime number greater than three, 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. Ianucci 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 that are multiples of powers of 3 greater than 9 but not themselves powers of three.

- 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. - Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L. (2003). "On perfect totient numbers".
*Journal of Integer Sequences*6 (4): 03.4.5. 2051959.

| url = http://www.emis.de/journals/JIS/VOL6/Cohen2/cohen50.pdf}}

- Luca, Florian (2006). "On the distribution of perfect totients".
*Journal of Integer Sequences*9 (4): 06.4.4. 2247943.

| url = http://www.emis.ams.org/journals/JIS/VOL9/Luca/luca66.pdf}}

- Mohan, A. L.; Suryanarayana, D. (1982). "Perfect totient numbers".
*Number theory (Mysore, 1981)*, 101–105. 0665442. .}} - Venkataraman, T. (1975). "Perfect totient number".
*The Mathematics Student*43 178. 0447089. }}

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