Is there any formula to calculate: $$\lfloor n\rfloor+\left\lfloor \frac n2\right\rfloor+ \left\lfloor \frac n3\right\rfloor+\ldots+\left\lfloor \frac nk\right\rfloor$$ with $n$ and $k$ positive integers and $k\leq n$.

  • $\begingroup$ Is $[\cdot]$ the integer part? $\endgroup$ – marco trevi Sep 30 '14 at 10:36
  • $\begingroup$ @marcotrevi: Yes. $\endgroup$ – mac Sep 30 '14 at 10:37
  • 2
    $\begingroup$ I doubt there is a nice formula for this. There are two regimes: as long as $k^2<n$, the quotients are strictly decreasing with decreasing steps; the sums will approximately follow an $n\log(n)$ curve; then the quotients start repeating longer and longer, with a unit step in between. The last "run" goes for $n/2\le k\le n$, with the sum linearly increasing. $\endgroup$ – Yves Daoust Sep 30 '14 at 11:05

It does not seem that there exists a closed formula. Indeed, let us denote the sum to be calculated by $S_{n,k}$. Specializing to the case $k=n$ corresponds to the entry A006218 at OEIS, which only provides asymptotic estimates: $$n \left(\ln n + 2 \gamma - 1\right) - 4 \sqrt n - 1 \leq S_{n,n} \leq n \left(\ln n + 2 \gamma - 1\right) + 4 \sqrt n.$$

  • $\begingroup$ oeis.org/A006218 is not really the same thing, is it? The OP mentions several functions of $n$, one for each $k$. $\endgroup$ – lhf Sep 30 '14 at 11:44
  • $\begingroup$ @Ihf indeed, I will edit accordingly. Thanks! $\endgroup$ – Start wearing purple Sep 30 '14 at 11:49
  • $\begingroup$ @O.L.: what is $\gamma$? $\endgroup$ – mac Sep 30 '14 at 11:57
  • $\begingroup$ @mac Euler-Mascheroni constant. $\endgroup$ – Start wearing purple Sep 30 '14 at 11:58

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