Any n-digit number that is the sum of the n-th powers of its digits is called narcissistic. Four three-digit examples are 153, 370, 371, and 407. To illustrate, 1^3 + 5^3 + 3^3 = 153. There are, excluding zero, 88 of these integers. One might suppose that any other n-digit number is necessarily altruistic, in the sense that the sum of the n-th powers of its digits is a number other than itself. However, if we repeat the process with the new number, yielding a third, and that a fourth, and so on, it would appear that we always eventually end up with either a narcissistic number or in a cycle wherein each integer serves ultimately to regenerate itself. These cyclists are really just self-serving altruists: Let's call them pseudo-altruistic so as to distinguish them from either narcissistic (period-1 cyclists) or the truly altruistic numbers that, once placed into the iteration machinery, will not reoccur.

Eric Angelini suggested the iteration transform ten days ago and wondered, as did I, what pseudo-altruistic numbers are out there. To (at least partially) answer that, I have created a program that will determine all twenty-digits-or-less cyclists: One of the pleasures of programming (I use Mathematica) is taking a relatively simple construction (such as the one that generated the 51 brute-force-derived cycles on Eric's page) and transforming it into the finely-tuned engine that is, at this moment, engaged in the search. Yes I can!