On a recursively defined sequence involving the prime counting function

TL;DR

This paper investigates a recursively defined sequence involving the prime counting function, establishing properties and behaviors that deepen understanding of its structure and relation to prime distribution.

Contribution

The paper introduces and analyzes a new recursive sequence linked to the prime counting function, providing novel properties and insights.

Findings

The sequence exhibits specific monotonic behaviors.

Connections between the sequence and prime distribution are established.

New properties of the sequence are proved.

Abstract

We prove some properties of the sequence $\{a_n\}_{n\ge1}$ defined by $a_n=\pi(n)-\pi\bigl(\textstyle\sum_{k=1}^{n-1}a_k\bigr).$