Can $\sum_{k=1}^nk!$ be a perfect number?
I think an odd perfect number can be divisible by $9$ and the last digit of an odd perfect number can be $3$, so I think the above sum can be a perfect number.
Am I correct?
2026-03-26 13:44:39.1774532679
Can $1!+2!+3!+\cdots n!$ be a perfect number?
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No, such numbers cannot be perfect numbers. From the 22nd term onwards, the sum is divisible by $11$ but not by $121$, and odd perfect numbers need to have all prime factors $p \equiv 3 \bmod 4$ with even exponents.