On primes and practical numbers

TL;DR

This paper studies the distribution of practical numbers that are shifted primes, proves that large odd numbers are sums of a prime and a practical number, and explores conjectures related to practical numbers and prime tuples.

Contribution

It improves existing theorems on practical numbers and primes, proves a conjecture of Margenstern, and establishes bounds for practical number analogues of prime tuples.

Findings

Practical numbers that are shifted primes are more common than previously shown.

All large odd numbers can be expressed as the sum of a prime and a practical number.

The paper establishes the correct upper bound for an analogue of the prime k-tuples conjecture for practical numbers.

Abstract

A number $n$ is practical if every integer in $[1,n]$ can be expressed as a subset sum of the positive divisors of $n$. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and Weingartner. In addition, essentially proving a conjecture of Margenstern, we show that all large odd numbers are the sum of a prime and a practical number. We also consider an analogue of the prime $k$-tuples conjecture for practical numbers, proving the "correct" upper bound, and for pairs, improving on a lower bound of Melfi.