Powers of Sums of Digits
Sumit Mohanty, Hemant Kumar · Mathematics Magazine · 1979
Let G(A) be the sum of the squares of the digits in the integer A. Then, denoting by Gn(A) the result of n successive applications of the operator G to A, A. Porges proved (in [3]) that for any number A there exists either a positive integer n such that Gn(A) = 1 or a positive integer m such that Gm(A)=4. Since the set of eight numbers 4, 16, 37, 58, 89, 145, 42 and 20 is closed under the operation G, every natural number is eventually transformed by G either to unity or to one of these eight numbers. Kiyosi Iseki generalised Porges' problem in [2] showing that sums of cubes produce the following cyclic sequences: