DOI$^2$

TL;DR

This paper explores various intriguing properties of integers and their neighbors, revealing complex combinatorial, ordering, arithmetical, and probabilistic behaviors that challenge common assumptions.

Contribution

It presents new insights into the complex behaviors of integers and their neighbors, highlighting unexpected properties across multiple mathematical domains.

Findings

Neighbors of integers exhibit complex combinatorial properties

Integer sequences show surprising arithmetical and probabilistic behaviors

The observations challenge common expectations about integer properties

Abstract

We discuss some seemingly unrelated observations on integers, whose close or farther away neighbors show a complex of combinatorial, ordering, arithmetical or probabilistic properties, emphasizing puzzlement in more common expectations.