Here is a fairly computationally challenging 1994 AIME problem .

“Find the positive integer *n* for which

⌊log_{2} 1⌋ + ⌊log_{2} 2⌋ + ⌊log_{2} 3⌋ + … + ⌊log_{2} *n*⌋ = 1994.

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

There is some fussy consideration of indices.

See the Special Log Sum for a solution.