# Problem 6. (4 points)

Let $\sigma(n)$ denote the sum of all divisors of $n$ (including $n$ itself). For which $n$ does the inequality $\sigma(8 n)>\sigma(9 n)$ hold?
# Solution:

For each prime factor $p$, such that $n$ is divisible by $p^k$ but not by $p^{k+1}$, the contribution to $\sigma(n)$ is $\left(1+p+\ldots+p^k\right)$. If we write this strictly, for $n=\prod_{i=1}^{m} p_i^{k_i}$, the equality $\sigma(n) = \prod_{i=1}^{m}\left(1+p_i+\ldots+p_i^{k_i}\right)$ holds.

From this, it is clear that when multiplying or dividing the number $n$ by some factor that is coprime with the numbers 8 and 9 (i.e., not divisible by 2 or 3), both sides of the inequality $\sigma(8n) > \sigma(9n)$ change by the same factor. Therefore, it is sufficient to find all solutions of the form $n=2^k 3^l$ - all other solutions can be obtained by multiplying these by some number $x$ that is not divisible by 2 or 3.

Then, $\sigma(8n) = \left(1+2+\ldots+2^{k+3}\right)\left(1+3+\ldots+3^l\right)$, and $\sigma(9n) = \left(1+2+\ldots+2^k\right)\left(1+3+\ldots+3^{l+2}\right)$. Their ratio should be greater than 1:

$$
\frac{\sigma(8n)}{\sigma(9n)} = \frac{\frac{1+2+4+8+\ldots+2^{k+3}}{1+2+\ldots+2^k}}{\frac{1+3+9+\ldots+3^{l+2}}{1+3+\ldots+3^l}} = \frac{\frac{7}{1+2+\ldots+2^k} + 8}{\frac{4}{1+3+\ldots+3^l} + 9} > 1
$$

The denominator of this fraction is always greater than 9. The numerator equals 9 when $k=2$ and is less than 9 for larger $k$. It remains to consider two cases: $k=1$ and $k=0$.

For $k=1$, we have:

$$
\begin{aligned}
& \frac{7}{3} + 8 > \frac{4}{1+3+\ldots+3^l} + 9 \\
& 10 \frac{1}{3} > 9 + \frac{4}{1+3+\ldots+3^l}
\end{aligned}
$$

The right-hand side equals 13 when $l=0$. For $k \geq 1$, the right-hand side is no more than 10. Therefore, $\sigma(8n) > \sigma(9n)$ when $k=1$ and $l \geq 1$.

For $k=0$, we have:

$$
15 > 9 + \frac{4}{1+3+\ldots+3^l}
$$

This inequality is always true. Thus, we have two series of answers:

Answer: $n = 3^l x$ or $n = 6 \cdot 3^l x$, where $l$ is a non-negative integer, and $x$ is a natural number not divisible by 2 or 3.