The integer recurrence P(n)=a+P(n-phi(a)) I

TL;DR

This paper proves that a specific integer recurrence relation generates all integers coprime to a in increasing order and provides its Fourier expansion and properties.

Contribution

It introduces a new recurrence relation involving Euler's totient function that systematically generates coprime integers and analyzes its properties.

Findings

Sequence generates all coprime integers in increasing order

Finite Fourier expansion of the sequence is derived

Properties of the sequence are characterized

Abstract

We prove that for a positive integer a the integer sequence P(n) satisfying for all n, -infty<n<infty, the recurrence P(n)=a+P(n-phi(a)), phi(a) the Euler function, generates in increasing order all integers P(n) coprime to a.The finite Fourier expansion of P(n) is given in terms of a, n, and the phi(a)-th roots of unity. Properties of the sequence are derived.