Cfrgtkky

up vote 19 down vote favorite 7 FINAL EDIT AND SUMMARY. The basis of this problem, and that which allows for the approximations to be made here, can be summarised in one approximation: $$Biggl(frac{n^k -{lfloor n^{frac{1}{k}} rfloor}^{k-1}gcd({lfloor n^{frac{1}{k}} rfloor}^{k-1},Bigllfloor frac{p_n^{k-1}}{n^{k-1}} Bigrrfloor)}{n^k -{lfloor n^{frac{1}{k}} rfloor}gcd({lfloor n^{frac{1}{k}} rfloor},Bigllfloor frac{p_n^{k}}{n^{k}} Bigrrfloor)}Biggr)^{frac{1}{k}} sim1quadforall n,k in mathbb Nbackslash {{1}}$$ $$quadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquadquad(A0)$$ $$frac{Bigllfloor bigl(n^k -{lfloor n^{frac{1}{k}} rfloor}^{k-1}gcd({lfloor n^{frac{1}{k}} rfloor}^{k-1},Bigllfloor frac{p_n^{k-1}}{n^{k-1}} Bigrrfloor)bigr)^{frac{1}{k}}Bigrrfloor }{Bigllfloor bigl(n^