(Note: This question has been cross-posted to MO.)

Euclid and Euler proved that every even perfect number is of the form $m = \frac{{M_p}\left(M_p + 1\right)}{2}$ where $M_p = 2^p - 1$ is a prime number, called a Mersenne prime. Thus, an even perfect number is triangular.

On the other hand, Euler showed that an odd perfect number, if one exists, takes the form $N = q^k n^2$, where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. (Descartes, Frenicle and subsequently Sorli conjectured that $k = 1$ always holds.)

Here is my question:

Has it been proved that odd perfect numbers cannot be triangular?

Added March 26 2016

If $\sigma(q) = 2n^2$, then it would follow that $n < q$, which implies that $k = 1$. The odd perfect number $N = q^k n^2$ then takes the form $N = \frac{q(q + 1)}{2}$. Unfortunately, it is known that $\sigma(q^k) \leq \frac{2n^2}{3}$.

Any pointers to the existing literature containing such a proof would be most appreciated.


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