perfect totient number

An integer $n$ is a perfect totient number if

 $n=\sum_{i=1}^{c+1}\phi^{i}(n)$

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

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 $p\equiv 3\mod 4$. 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 $x>3$.

References

• 1 Perez Cacho, “On the sum of totients of successive orders,” Revista Matematica Hispano-Americana 5.3 (1939): 45 - 50
• 2 D. E. Ianucci, D. Moujie & G. L. Cohen, “On Perfect Totient Numbers” Journal of Integer Sequences, 6, 2003: 03.4.5
• 3 R. K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: B42
Title perfect totient number PerfectTotientNumber 2013-03-22 16:33:20 2013-03-22 16:33:20 CompositeFan (12809) CompositeFan (12809) 6 CompositeFan (12809) Definition msc 11A25 totient perfect number IteratedTotientFunction