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Fix positive integers $m,n$. Consider values that can be expressed as a sum

$$x_1 + {x_2 \over 2} + {x_3 \over 3} + \cdots + {x_n \over n}$$

where each $x_i$ ranges over the set $\{0,1,2,3,\dots,m\}$. How many different such values are there?

Let $f(m,n)$ denote the number of such values -- I'm wondering if I can get a crude asymptotic estimate for $f(m,n)$. The application is for an algorithm in computer science, so I'm most interested in whether it grows exponentially fast or polynomially fast. In particular, for fixed constant $c_1$, assuming $m \le n^{c_1}$, does $f(m,n)$ grow exponentially fast (there is $c_2$ such that $f(n^{c_1},n) \ge e^{c_2 n}$ for sufficiently large $n$)? I suspect it does but can't see how to approach the problem.

D.W.
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In the case $m \ge n$, I can prove that $f(m, n) \ge \prod_{p\le n} p$, where the product is only over primes $p \le n$. Using the prime number theorem, we can show that this lower bound is on the order of $e^n$: see this MSE post. Wikipedia also has some citations for the asymptotics of the Chebyshev function $\vartheta(n)$, which is the log of this bound.

To prove this, I'll restrict the sums to cases where $x_i = 0$ when $i$ is not prime, and $x_p < p$ when $p$ is prime: since $m \ge n$, all values of $x_p$ between $0$ and $p-1$ are still allowed. There are $p$ possibilities for $x_p$, giving the product above; now I'll prove that all resulting sums are different.

Given a prime $p \le n$, we can write any sum of this form as $\frac ab + \frac cp$ where $p \nmid b$, and if we had $\frac ab + \frac cp = \frac{a'}{b'} + \frac{c'}{p}$, then $p \left(\frac ab - \frac{a'}{b'}\right)= c-c'$. Unless the right-hand side is $0$, it is an integer between $1$ and $p-1$, so it is not divisible by $p$; the left-hand side is either not an integer or divisible by $p$ (since $p$ can't help cancel fractions in $\frac ab - \frac{a'}{b'}$). Therefore these can only be equal if $c=c'$.

This means that in any two equal sums of this form, we must have the same value of $x_p$. This holds for all $p$, so all coefficients must be identical.

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Misha Lavrov
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