TH-TMO 2005 Problem 10

What is the remainder when $\sum_{k=1}^{2005}k^{2005\cdot 2^{2005}}$ is divided by $2^{2005}$ ?

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.