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arXiv · 2508.06651

Complete characterization of $2$-near perfect numbers with exactly 2 prime factors

Richard Fearon·Henry Foushee·Benjamin Porosoff·Alexander Skula·Joshua Zelinsky·Kyle Zhang
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Abstract

Let $σ(n)$ be the sum of the positive divisors of $n$. A positive integer $n$ is said to be $2$-near perfect when $σ(n)=2n+d_1+d_2$, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We show that there are no odd $2$-near perfect numbers with exactly two prime factors, and that all even $2$-near perfect numbers (i.e. those of the form $2^kp^m$, where $p$ is an odd prime) belong to a specific family, provided that $m$ is at least 3. In combination with prior work, these results produce a complete characterization of $2$-near perfect numbers with exactly 2 prime factors.

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Richard Fearon, Henry Foushee, Benjamin Porosoff, Alexander Skula, Joshua Zelinsky, Kyle Zhang. 2026-05-23. Complete characterization of $2$-near perfect numbers with exactly 2 prime factors. https://arxiv.org/abs/2508.06651

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