THE INTEGER RECURRENCE P(n)=a+P(n-φ(a))

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.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 33
Issue: 3
Year: 2020

DOI: 10.12732/ijam.v33i3.2

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