Sums of Functions of Digits

Extract

We generalize in several directions a paper by Porges (2) who considered the integer F(A) obtained from the positive integer .1 by taking the sum of the squares of the digits of A. Porges showed that if A > 99, then F(A) < A, so that under iteration of F(A) all the positive integers are divided into a finite number of classes, called orbits in the terminology of Isaacs (1), each containing a finite cycle. For his F(A) Porges showed there are only two orbits: one with the 1-cycle: 1 → 1 ; and the other with the interesting 8-cycle: 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.

Consider the set Z of non-negative integers and choose as a base of enumeration any desired integer B ≧ 2 (not necessarily B = 10). Then only the “digits” 0, 1, 2, … , B — 1 are needed, in suitable multiplicity, to represent any A of Z.