p[i] is the i-th prime. $\pi(x)$ is prime counting function.
Firstly, I think that this Prime gap inequality holds true,
$ p[i+1] - p[i] <= i $
Prove:for any i>0, we can always find distinct prime factors for {p[i], p[i]+1,...,p[i+1]}. For example, i=11, p[11]=31, p[12]=37, {31,32,33,34,35,36,37} have distinct prime factors {31,2,11,17,5,3,37}. Pigeonhole principle shows this simple inequality!
My question is the title,
Conjecture(Prime counting inequality) :
if $ i\lt j $, then $\pi(p[i]+i)-i<=\pi(p[j]+j)-j+1$
Edit: sorry, this conjecture is false! But another question is arising: if $ i\lt j $, what is the max value of $g(i,j) = (\pi(p[i]+i)-i) - (\pi(p[j]+j)-j) /; i\lt j$
I find g(i,j) may be 12. g(150065,150090)=12.
and more, what is the max value of h(i,j)=j-i /; $(\pi(p[i]+i)-i) > (\pi(p[j]+j)-j) $
