Problem of the Day #80

POTD February 26, 2024

Consider an integer 0 \le n \le 127. We define 3 operations:

1. If n \le 63, we can replace n with 2n.

2. If n \ge 64, we may replace n with 2n - 127.

3. If 95 \ge n \ge 64, we may replace n with n - 32.

Define f(n) as the number of integers n can reach with 0 or more operations. Find the sum of all f(n) across all 0 \le n \le 127.