1, 2, 4, 16, 26, 42, 57, 512, 730, 1010, 1343, 1872, 2367, 2954

I noticed the 4th term is 4^2 and the 8th term is 8^3, also the 1st term is 1^0 and the 2nd is 2^1

Maybe each is just the term number raised to its base two logarithm?  This would mean f(n) = n^(log(n)/log(2))

Nope.  f(3) = 5.7045 and f(5) = 41.9718  Although if it hold for powers of two the 16th term is 65536 and I am only looking for the 15th term which by inspection is around 3500.

So maybe the powers of n are important but they follow some other pattern

The needed powers are 0, 1, 1.2618, 2, 2.024, 2.086, 2.078, 3, 3.0056, 3.0043, 3.0044, 3.0322, 3.0291, 3.0279, ?, 4?

I think I am close.  Those jumps at the powers of two are intreguing.  Also odd is the fact that this list is not strictly increasing.  It also has a mini-jump from 11 to 12.

I'm not finished with this one yet...

-Jer