The number of multinomial coefficients based on a set of partitions of n into k parts and divided by k evenly

TL;DR

This paper introduces a new integer sequence derived from multinomial coefficients related to partitions of n, revealing a connection to primality testing and proposing a potential asymptotic primality algorithm.

Contribution

It presents a novel sequence based on multinomial coefficients and explores its properties, linking it to primality testing and suggesting an asymptotic primality algorithm.

Findings

For prime n, the sequence's n-th term is one less than the number of partitions of n.

The sequence exhibits properties that relate to primality.

A hypothesis is proposed for an asymptotic primality testing algorithm.

Abstract

In this paper we obtained an original integer sequence based on the properties of the multinomial coefficient. We investigated a property of the sequence that shows connection with a primality testing. For any prime n the n-th term in the sequence is less by 1 than the number of partitions of n. We hypothesize the existence of an asymptotic algorithm of primality testing.