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

2025/08/08 by Fearon, Richard, Foushee, Henry, Porosoff, Benjamin +3
#11A25 #11N25 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.06651

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+d₁+d₂, where d₁ and d₂ 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.

Citations

• On 2-Near Perfect Numbers
• On near-perfect numbers of special forms
• On perfect and near-perfect numbers

Related

• On 2-Near Perfect Numbers
• New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number
• On Exactly 3-Deficient-Perfect Numbers
• On the small prime factors of a non-deficient number
• On odd deficient-perfect numbers with four distinct prime divisors