Dynamics of a function related to the primes

Abstract

Let n = p ₁ p ₂ ⋯ p _k, where p _i (1 ≤ i ≤ k) are primes in the descending order and are not all equal. Let Ωk(n) = P(p ₁+p ₂)P(p ₂+p ₃) ⋯ P(p k−1+p _k)P(p _k +p ₁), where P(n) is the largest prime factor of n. Define w ⁰(n) = n and w ⁱ(n) = w(w ⁱ⁻¹(n)) for all integers i ≥ 1. The smallest integer s for which there exists a positive integer t such that Ω _k ^s(n) = Ω _k ^s+t(n) is called the index of periodicity of n. The authors investigate the index of periodicity of n.

Keywords

Dynamics / The largest prime factor / Arithmetic function

References

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