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 (n+1) (n+2) (n+k) holds for some n x, where is the Euler totient function. Their method of proof also works to derive an upper bound for k with replaced by, 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 (n+1) = (n+2) == (n+k). Here we improve their bound with a similar method of proof.