When π(n) does not divide n

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Let π(n) denote the prime-counting function and let f(n)=| n- n-0.1| n/ n-1 n-1/n. In this paper we prove that if n is an integer ≥ 60184 and f(n)=0, then π(n) does not divide n. We also show that if n≥ 60184 and π(n) divides n, then f(n)=1. In addition, we prove that if n≥ 60184 and n/π(n) is an integer, then n is a multiple of n-1 located in the interval [e n-1+1,e n-1+1.1]. This allows us to show that if c is any fixed integer ≥ 12, then in the interval [eᶜ,eᶜ⁺⁰.¹] there is always an integer n such that π(n) divides n. Let S denote the sequence of integers generated by the function d(n)=n/π(n) (where n and n>1) and let Sₖ denote the kth term of sequence S. Here we ask the question whether there are infinitely many positive integers k such that Sₖ=Sₖ₊₁.