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Streszczenie

Let $\sigma_{10,j}(n)$ be the sum of all positive divisors $d$ of $n$ which satisfy $d\equiv j$ (mod $10$). We deal with the conjecture that $\sigma_{10,1}(n)\neq \sigma_{10,9}(n)$ for any positive integer $n$. Our result is that the conjecture holds for $n\equiv \pm 3$ (mod $10$) and also for all $n$ with $\omega^\prime (n)\leq 60\,060$, where $\omega^\prime (n)$ stands for the number of distinct primes dividing $n$, which are of the form $10k\pm 3$. In particular, the conjecture holds for all $n \lt 10^{340000}$.