On Some Results on Practical Numbers

TL;DR

This paper investigates the distribution of practical numbers within polynomial sequences, providing conditions for their infinitude, solving a related conjecture, and exploring their additive properties with squares.

Contribution

It establishes necessary and sufficient conditions for infinite practical numbers in linear and quadratic forms and addresses conjectures about their additive representations.

Findings

Infinite practical numbers exist in certain polynomial forms based on coefficient conditions.

A quadratic polynomial can contain infinitely many practical numbers under specific criteria.

Every number of the form 8k+1 can be expressed as a sum of a practical number and a square.

Abstract

A positive integer $n$ is said to be a practical number if every integer in $[1,n]$ can be represented as the sum of distinct divisors of $n$. In this article, we consider practical numbers of a given polynomial form. We give a necessary and sufficient condition on coefficients $a$ and $b$ for there to be infinitely many practical numbers of the form $an+b$. We also give a necessary and sufficient for a quadratic polynomial to contain infinitely many practical numbers, using which we solve first part of a conjecture mentioned in [9]. In the final section, we prove that every number of $8k+1$ form can be expressed as a sum of a practical number and a square, and for every $j\in \{0,\ldots,7\}\setminus \{1\}$ there are infinitely many natural numbers of $8k+j$ form which cannot be written as sum of a square and a practical number.