Let b ≥ 2 be an integer and g = b - 1. We consider a generalization of the modified Collatz function: for any positive integer m, the g-Collatz function f_g divides m by g, if m is a multiple of g; otherwise, the g-Collatz function f_g is the least integer greater than or equal to bm/g . Using this g-Collatz function, we extend the Collatz problem, and we show that there are nontrivial cycles for some g. Then we show how the function f_g transforms the base-b representation of positive integers, and we study the sequence of the b-ary representation of integers generated by the function f_g, starting with a b-ary string representing b^N for an arbitrary large integer N. We show each b-ary string in the sequence has a repeating string, and the number of occurrences of each digit in each shortest repeating string generalizes Jacobsthal numbers.