Perfect totient numbers (nonfiction)
In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients.
That is, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers; if the sum equals n, then n is a 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 9 = 6 + 2 + 1, 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, ... (sequence A082897 in the OEIS).
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 ...
In the News
Fiction cross-reference
Categories
Nonfiction cross-reference
- Euler's totient function (nonfiction)
- Gnomon Chronicles (nonfiction)
- Mathematical induction (nonfiction)
Categories
External links
- Mathematics @ Wikipedia
Social media
- [ Post] @ Twitter (27 November 2025)
