Subnormal subgroup

There are functions μ1, μ2: ℕ → ℕ, such that any, all of whose , which can be generated by μ2(n) elements, are n-step , is nilpotent of class ≤ μ1(n). It is defined by:


 * μ₁(1) = 2
 * μ₁(n) = μ₆(μ₇(n)) (n>1)
 * μ₂(1) = 1
 * μ₂(n) = μ₂(n-1) + μ₇(n) (n>1)
 * μ₄(n,1) = 1
 * μ₄(n,2) = 1 + 2ⁿ⁻¹ + μ₉(n, μ₈(n))
 * μ₄(n,m) = μ₁₂(μ₄(n,m-1), μ₄(n,2)) (m>2)
 * μ₅(n) = n μ₁₃(n)
 * μ₆(n) = μ₄(n, μ₅(n))
 * μ₇(n) = 1 + μ₄(n, (n-1) μ₁(n-1)) (n>1)
 * μ₈(1) = 1
 * μ₈(n) = n (μ₈(n-1))² (n>1)
 * μ₉(1,m) = 2
 * μ₉(n,m) = n - 1 + μ₁₁(n,m) (n>1)
 * μ₁₀(n) = n + n² + … + nn²
 * μ₁₁(n,m) = ⌊μ₁₀(n) log₂(m)⌋ + 1
 * μ₁₂(c,d) = (c+1)cd/2 - c(c-1)/2
 * μ₁₃(n) = n + μ₁₅(n)
 * μ₁₄(n) = n + n² + … + nⁿ
 * μ₁₅(n) = μ₁₆(μ₁₄(n)) + 1
 * μ₁₆(n) = ½n(5n-1) + 1