For n>1, A108309 is related to the number A(n) of primes
between n^2-n and n^2+n.
Using Prime Number Theorem, it is easy to prove that the asymptotic mean
of A(n) is equivalent to n / log(n) with log(e)=1.
Now we conjecture that :
the gap G(n) between A(n) and n/log(n) is always less than sqrt(n) with
sqrt(n) = square root of n.
For example :
for n=37 , A(n)=5 , n/log(n)=10,2467 and G(n) / sqrt(n) =
5,2467/sqrt(37) = 0,86255.
The next largest ratio is obtained for n=5123 :
for n=5123 , A(n)=538 , n/log(n)=599,7779 and G(n) / sqrt(n) =
61,7779/sqrt(5123) = 0,86312.
Q1. Using an extension of A108309, can you get
the values of n leading to larger ratios?
Q2. Can you get an explanation of this conjecture?
