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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. Or to put it algebraically, if
$n = sum_\left\{i = 1\right\}^\left\{c + 1\right\} varphi^i\left(n\right),$
where
$varphi^i\left(n\right)=left\left\{begin\left\{matrix\right\}varphi\left(n\right)&mbox\left\{ if \right\} i=1 varphi\left(varphi^\left\{i-1\right\}\left(n\right)\right)&mbox\left\{ otherwise\right\}end\left\{matrix\right\}right.$
is the iterated totient function and c is the integer such that
$displaystylevarphi^c\left(n\right)=2,$
then n 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, ... .

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.

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

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

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

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.

References

• 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.

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• 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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