Perfect totient number

Examples

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, ... .

Notation

In symbols, one writes :\varphi^i(n) = \begin{cases} \varphi(n), &\text{ if } i = 1 \ \varphi(\varphi^{i-1}(n)), &\text{ if } i \geq 2 \end{cases} for the iterated totient function. Then if c is the integer such that :\displaystyle\varphi^c(n)=2, one has that n is a perfect totient number if :n = \sum_{i = 1}^{c + 1} \varphi^i(n).

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 :\displaystyle\varphi(3^k) = \varphi(2\times 3^k) = 2 \times 3^{k-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 3k**p where p is prime and k 3.

References

• {{cite journal

• {{cite book | author-link = Richard K. Guy

• {{cite journal | access-date = 2007-02-07 | archive-date = 2017-08-12 | archive-url = https://web.archive.org/web/20170812121811/http://www.emis.de/journals/JIS/VOL6/Cohen2/cohen50.pdf | url-status = dead

• {{cite journal | author-link = Florian Luca | access-date = 2007-02-07

• {{cite conference | book-title = Number theory (Mysore, 1981)

• {{cite journal

• {{cite web