Some Consequences of Bertrand's Extended Postulate

Bertrand's postulate establishes that for all positive integers n > 1 there exists a prime number between n and 2n. We consider a generalization of this theorem as: for integers n ≥ k ≥ 2 is there a prime number between kn and (k + 1)n? This is a generalization of Bertrand's postulate extended as proved at link 1706.01009.pdf. The example is deduced that there are at least k -1 prime numbers between n and kn where n, k is a positive integers greater than 1. Then we can prove a number of hypotheses and some properties below. And here are the consequences to be deduced from it.