Would you please help me prove or disprove the conjecture highlighted below. I conceived it after solving a textbook problem to find distinct $n_1, n_2$ such that $n_1 \sigma(n_1) = n_2 \sigma(n_2)$ where $\sigma$ is the sum-of-divisors function. Out of curiosity, I used a table of the first one hundred thousand values of $\sigma$ and found 3408 such pairs: 12, 14; 48, 62; . . .; 94860, 95472. Out of more curiosity, I found 96 triples such that $n_1 \sigma(n_1) = n_2 \sigma(n_2) = n_3 \sigma(n_3)$: 336, 372, 434; . . .; 85680, 94860, 95472. Continuing, I found two quadruples such that $n_1 \sigma(n_1) = n_2 \sigma(n_2) = n_3 \sigma(n_3) = n_4 \sigma(n_4)$: 41664, 42672, 47244, 55118 and 42000, 46500, 51200, 54250.
A natural conjecture, then, is
For any positive integer $r$, there exist distinct positive integers $n_1, n_2, \dots, n_r$ such that $n_1 \sigma(n_1) = n_2 \sigma(n_2) = \cdots = n_r \sigma(n_r)$.
If the conjecture is true, then my search failed for $r \ge$ 5 only because my list of $\sigma$ values was not large enough.