Special Log Sum

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.

See the Special Log Sum.