perfect totient number

An integer n is a perfect totient number if


, where ϕi(x) is the iterated totient function and c is the integer such that ϕc(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 multiplesMathworldPlanetmath of 3 (in fact, 4375 is the smallest one that is not divisible by 3) and in fact all 3x 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 p3mod4. In regards to 32p, 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 3xp for x>3.


  • 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 TheoryMathworldPlanetmathPlanetmath New York: Springer-Verlag 2004: B42
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