Problem of the Day #48

POTD January 22, 2024

Let f(S) be the number of pairs of consecutive numbers in the set S. For example,

[f \left(\{\mathbf{1}, \mathbf{2}, 5, 7, \mathbf{9}, \mathbf{10}, 12\} \right) = 2.

For a random subset T \subseteq \{1,2,\ldots,2024\} such that |T| = k, the expected value of f(T) is E(k). Find the number of factors of \sum_{i=0}^{2024} E(i).