Joshua Zelinsky

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20242026

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math.NT2026

Must a primitive non-deficient number have a component not much larger than its radical?

Joshua Zelinsky

Let $n$ be a primitive non-deficient number where $n = p_{1}^{a_{1}} p_{2}^{a_{2}} \dots p_{k}^{a_{k}}$ where $p_{1}, p_{2}, \dots, p_{k}$ are distinct primes. We prove that there exists an $i$ such…

math.NT2026

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

Richard Fearon, Henry Foushee, Benjamin Porosoff +3

Let $I ¨ (n)$ be the sum of the positive divisors of $n$ . A positive integer $n$ is said to be $2$ -near perfect when $I ¨ (n) = 2 n + d_{1} + d_{2}$ , where $d_{1}$ and $d_{2}$ are distinct positive d…

math.NT2025

On near superperfect numbers, the Goormaghtigh conjecture, and Mertens' theorem

Satvik Beri, Joshua Zelinsky

Let $I ¨ (n)$ be the sum of the divisors of $n$ . Kalita and Saikia defined a number $n$ to be near superperfect if $2 n + d = I ¨ (I ¨ (n))$ for some positive divisor $d$ of $n$ . We extend so…

math.NT2025

A Note on a Result of Makowski

Luis H. Gallardo, Joshua Zelinsky

In this note, we fix a gap in a proof of the first author that 28 is the only even perfect number which is the sum of two perfect cubes. We also discuss the situation for higher po…

math.NT2025

The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number

Joshua Zelinsky

Write $T (n)$ as the sum of the reciprocals of the primes which divide $n$ . Write $H (n) = \prod_{p ∣ n} p / (p - 1)$ where the product is over the prime divisors of $n$ . We prove new bound…

math.NT2025

Kullback-Leibler divergence and primitive non-deficient numbers

Joshua Zelinsky, Kyle Zhang

Let $H (n) = \prod_{p ∣ n} \frac{p}{p - 1}$ where $p$ ranges over the primes which divide $n$ . It is well known that if $n$ is a primitive non-deficient number, then $H (n) > 2$ . We exami…

Must a primitive non-deficient number have a component not much larger than its radical?

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

On near superperfect numbers, the Goormaghtigh conjecture, and Mertens' theorem

A Note on a Result of Makowski

The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number

Kullback-Leibler divergence and primitive non-deficient numbers

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