perfect totient number
, where is the iterated totient function and is the integer such that .
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 for are perfect totient numbers.
Furthermore, for a prime is a perfect totient number if and only if , where itself is also a perfect totient number. Mohan and Suryanarayana showed why can’t be a perfect totient number when . In regards to , Ianucci et al showed that if it is a perfect totient number then 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 for .
- 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|
|Date of creation||2013-03-22 16:33:20|
|Last modified on||2013-03-22 16:33:20|
|Last modified by||CompositeFan (12809)|
|Synonym||totient perfect number|