**odd prime**. We want to calculate $I(p)=p^{-1}\mod 2^p$.

If we use EXGCD, it takes $O(\log_22^p)=O(p)$ time, which is not efficient enough.

Some small values are 3, 13, 55, 931, 3781, 61681, 248347, 4011943, 259179061......

It seems that $I(p)$ is slightly smaller than $2^{p-1}$, but I didn't find more explicit patterns, nor did I find the sequence on OEIS.

So is there a closed form of $I(p)$, or at least can we calculate that more efficiently?

Any idea would be appreciated!