Arithmetic functions that remain constant on strings of integers

In 2013, Pollack et al. gave an asymptotic expression for the largest k in terms of x for which the string of inequalities φ(n+1)≥φ(n+2)≥⋯≥φ(n+k) holds for some n≤ x, where φ is the Euler totient function. Their method of proof also works to derive an upper bound for k with φ replaced by σ, the sum-of-divisors function. In 2023, the first author and Vandehey gave an alternative proof for a weaker lower bound on k in the string of equalities φ(n+1)=φ(n+2)=…=φ(n+k). Here we improve their bound with a similar method of proof.