Let a ≥ 2 be a fixed integer. Given a positive integer n, first divide n by a repeatedly until the resulting integer is no longer divisible by a. Denote the resulting integer by m. Next, multiply m by a² − 1. Then add a unique integer k chosen from the set 1, 2,. . . , a − 1 so that the resulting quantity becomes divisible by a. Finally, divide by a and repeat the procedure. For each integer m in the set 1, 2,. . . , a − 1, the iteration maps m to a multiple of a that returns to m after the removal of powers of a. Therefore each such m generates a trivial cycle. The special case a = 2 reproduces the accelerated Collatz map.
ali rismanchi (2026) studied this question.