On a sequence involving sums of primes

TL;DR

This paper investigates the behavior of sequences derived from sums of primes, proving monotonicity properties and proposing related conjectures involving twin primes and partitions.

Contribution

It establishes new monotonicity results for sequences based on prime sums and introduces conjectures connecting these sequences to twin primes and integer partitions.

Findings

Sequence a_n = (S_n/n)^(1/n) is strictly decreasing.

Sequence a_{n+1}/a_n is strictly increasing for n ≥ 10.

Formulation of conjectures involving twin primes and partitions.

Abstract

For $n=1,2,3,\ldots$ let $S_n$ be the sum of the first $n$ primes. We mainly show that the sequence $a_n=\root n\of{S_n/n}\ (n=1,2,3,\ldots)$ is strictly decreasing, and moreover the sequence $a_{n+1}/a_n\ (n=10,11,\ldots)$ is strictly increasing. We also formulate similar conjectures involving twin primes or partitions of integers.