Is there some mathematical closed form (or somewhat tight asymptotic one) for "Google Eggs Puzzle"? The following brief description of the known "Google Eggs Puzzle" comes mainly from the web site Google Eggs:


  Google Eggs Puzzle: Given n floors and m eggs, what is the approach to find the highest floor from which eggs can be thrown safely, while minimizing the throws (not broken eggs). 


The so called "highest floor" in the above problem deserves more formal definition:


  "highest:" there must be a floor f (in any sufficiently tall building) such that an egg dropped from the  f th floor breaks, but one dropped from the (f-1)st floor will not. Then, f-1 here is the highest floor.


Actually, the description of "highest" is an excerpt from the book "The Algorithm Design Manual (Second Edition)" by Steven S. Skiena. Being an exercise in Chapter 8 "Dynamic Programming", there are plenty of resources in Web devoted to solving the puzzle by the means of dynamic programming, like Google Eggs and The Two Egg Problem.

However, there is a question from the above book:


  Show that $E(n, m) = \Theta(n^{\frac{1}{m}})$, where $E(\cdot)$ is the minimum number of throws. (Note: I have changed the notations used in book for consistency.)


It is the question that motivates my problem:


  My Problem: Is there some mathematical closed form for general "Google Eggs Puzzle" with n floors and m eggs, instead of dynamic programming recurrence, and of course tighter than the $E(n, m) = \Theta(n^{\frac{1}{m}})$ one?
With m eggs and k measurements the most floors that can be checked is exactly $$n(m,k)={k \choose 0} + {k \choose 1} + \ldots + {k \choose m},$$ (maybe $\pm 1$ depending on the exact def). Proof is trivial by induction. This expression has no closed form inverse but gives good asymptotic.