Exceptional sets for compositions involving Euler's function,the divisor-sum function and Dedekind's function

Let \(ψ(n)\), \(ϕ(n)\), and \(σ(n)\) denote Dedekind's arithmetic function, Euler's totient function, and the sum-of-divisors function, respectively. We study exceptional sets associated with the compositions \(ϕ(ψ(n))\), \(ϕ(σ(n))\), \(ψ(ψ(n))\), and \(ψ(σ(n))\). For every fixed \(c>0\), we obtain