Here is a fairly computationally challenging 1994 AIME problem .

“Find the positive integer n for which

⌊log₂ 1⌋ + ⌊log₂ 2⌋ + ⌊log₂ 3⌋ + … + ⌊log₂ n⌋ = 1994.

where for real x, ⌊x⌋ is the greatest integer ≤  x.”

There is some fussy consideration of indices.

Answer

See the Special Log Sum for a solution.