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Consider ‘\’ to be the integer division operator, i.e.,

$a$ \ $b = \lfloor a / b\rfloor$

Is there a formula to compute the following summation:

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N\1 + N\2 + N\3 + … + N\N

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This is not a closed form, but an alternate characterization of this sum is

$$

\sum_{k=1}^n\lfloor n/k\rfloor=\sum_{k=1}^nd(k)\tag{1}

$$

where $d(k)$ is the number of divisors of $k$. This can be seen by noticing that $\lfloor n/k\rfloor$ increases by $1$ when $k\mid n$:

$$

\begin{array}{c|cc}

\lfloor n/k\rfloor&1&2&3&4&5&6&k\\

\hline\\

0&0&0&0&0&0&0\\

1&\color{#C00000}{1}&0&0&0&0&0\\

2&\color{#C00000}{2}&\color{#C00000}{1}&0&0&0&0\\

3&\color{#C00000}{3}&1&\color{#C00000}{1}&0&0&0\\

4&\color{#C00000}{4}&\color{#C00000}{2}&1&\color{#C00000}{1}&0&0\\

5&\color{#C00000}{5}&2&1&1&\color{#C00000}{1}&0\\

6&\color{#C00000}{6}&\color{#C00000}{3}&\color{#C00000}{2}&1&1&\color{#C00000}{1}\\

n

\end{array}

$$

In the table above, each red entry indicates that $k\mid n$, and each red entry is $1$ greater than the entry above it. Thus, the sum of each row increases by $1$ for each divisor of $n$.

A simple upper bound is given by

$$

n(\log(n)+\gamma)+\frac12\tag{2}

$$

This is because we have the following bound for the $n^\text{th}$ Harmonic Number:

$$

H_n\le\log(n)+\gamma+\frac1{2n}\tag{3}

$$

where $\gamma$ is the Euler-Mascheroni Constant.

**Research Results**

After looking into this a bit, I found that the Dirichlet Divisor Problem involves estimating the exponent $\theta$ in the approximation

$$

\sum_{k=1}^nd(k)=n\log(n)+(2\gamma-1)n+O\left(n^\theta\right)

$$

Dirichlet showed that $\theta\le\frac12$ and Hardy showed that $\theta\ge\frac14$.

There is no closed form known for $(1)$.

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