Given an integer $n\ge 1$, what is the largest eigenvalue $\lambda_n$ of the matrix $M_n=(m_{ij})_{1\le i,j\le n}$ with the elements $m_{ij}$ equal to $0$ or $1$ according to whether $ij>n$ or $ij\le n$?

It is not difficult to show that
  $$ c\sqrt n \le \lambda_n \le C\sqrt{n\log n} $$
for appropriate positive absolute constants $c$ and $C$, and numerical
computations seem to suggest that the truth may lie somewhere in between.