Ceil function equation I'm interested whether there is some standard way or any heuristic tricks to solve equations of the following type (N is unknown, $\alpha$ is positive irrational number, m is positive integer):
$$\sum_{k=1}^{N}\left \lceil{k ~\alpha}\right \rceil = m$$
For example, we're given a string of decimal digits like
Thanks in advance.

EDIT
In fact i'm interested in a way to solve a particular equation:
$$\sum_{k=1}^{N}\left \lceil{\log_{10}2^k}\right \rceil = m$$
but it will be very nice to have some general approach.
 A: You can get an initial guess for $N$ by using the formula for sum of consecutive integers, and drop the ceiling function, and divide both sides by $\alpha$, and solve for $N$ as a real number in terms of $m$ and $\alpha$, given the formula for sum of consecutive integers. Then round $N$ to the nearest integer. From there, probably your best bet is to do binary search starting at the initial value of $N$ after rounding: If your sum is too big, then start by subtracting $2^k$ from $N$ on the $k$th iteration until your sum is less than or equal to $m$.  If your sum is too small, then start by adding $2^k$ to $N$ on the $k$th iteration until your sum is greater than or equal to $m$. Then do binary search to find the exact value of $N$.
A: You can get a start by assuming the ceiling will add $1/2$ on average to each term.  This is justified for large $N$ by the equidistribution theorem.  Then you want $\sum_{k=1}^{N} (k ~\alpha+1/2) = \frac 12\alpha N(N+1)+\frac N2=m$  Solve the quadratic for $N$, then adjust $N$ up or down as required.
