2023 AIME II Problem 15

For each positive integer n let a_n be the least positive integer multiple of 23 such that a_n \equiv 1 \pmod{2^n}. Find the number of positive integers n less than or equal to 1000 that satisfy a_n = a_{n+1}.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.