(m,k)-perfect numbers

An integer n : ℤ is (m,k)-perfect if σᵐ(n) = kn where σᵐ is the mᵗʰ iterate of the sum of divisors function.

References:

• Wikipedia
• Wikipedia

def Superperfect.PerfectFor (n m k : ℕ) : Prop

An integer $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$ where $\sigma^m$ is the $m$-th iterate of $σ$.

Equations

• Superperfect.PerfectFor n m k = ((fun (x : ℕ) => (ArithmeticFunction.sigma 1) x)^[m] n = k * n)

theorem Superperfect.twoFivePerfect : ¬∃ (n : ℕ), PerfectFor n 2 5

There does not exist a $(2,5)$-perfect number