perfect totient number

An integer $n$ is a perfect totient number if

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

, where ${\varphi }^i(x)$ is the iterated totient function and $c$ is the integer such that ${\varphi }^c(n)=2$.

A082897 in Sloane’s OEIS lists the first few perfect totient numbers: 3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, etc. It can be observed that many of these are multiples of 3 (in fact, 4375 is the smallest one that is not divisible by 3) and in fact all $3^x$ for $x>0$ are perfect totient numbers.

Furthermore, $3p$ for a prime $p>3$ is a perfect totient number if and only if $p=4n+1$, where $n$ itself is also a perfect totient number. Mohan and Suryanarayana showed why $3p$ can’t be a perfect totient number when $p\equiv 3mod4$. In regards to $3^{2}p$, Ianucci et al showed that if it is a perfect totient number then $p$ is a prime of one of three specific forms listen in their paper. It is not known if there are any perfect totient numbers of the form $3^{x}p$ for $x>3$.

Title	perfect totient number
Canonical name	PerfectTotientNumber
Date of creation	2013-03-22 16:33:20
Last modified on	2013-03-22 16:33:20
Owner	CompositeFan (12809)
Last modified by	CompositeFan (12809)
Numerical id	6
Author	CompositeFan (12809)
Entry type	Definition
Classification	msc 11A25
Synonym	totient perfect number
Related topic	IteratedTotientFunction