Is there an equation for efficiency of a 2-radix FFT as you raise $k$?
No.

The FFT is pretty much an "all or nothing" algorithm. There is almost no efficiency gain in reducing the number of desired outputs. 

For $k \ll N$ the Goertzel algorithm is typically the better choice. See https://en.wikipedia.org/wiki/Goertzel_algorithm. The break even point is quite small. Maybe $k = 4\cdot log_2(N)$ or thereabouts.