Let a(n) be the largest number of consecutive integers such
that each is divisible by a prime <= the n-th prime.
On oeis.org : https://oeis.org/A058989
there is an upper bound mentioned which is
$ a(n) < $${n^2} \cdot {(ln(n))^2}$
This upper bound is approximately equal to $ {P_n}^2$
since $${P_n} > n \cdot \ln \left( n \right) $$
I wonder if this can be done better. My hope is that an upper bound can be found
where $ a(n) < \frac{{{n^2} \cdot {{(ln(n))}^2}}}{2}$
for some sufficiently large value of n.
A thought came to my mind.
Let a2(n) be the largest number of consecutive integers such
that each is divisible by an odd prime <= the n-th prime.
What would be the upper bound for this ?
If an upper bound for this can be shown to be less than
$ $${a_2}(n) < \frac{{{n^2} \cdot {{(ln(n))}^2}}}{4} $
then we would also have that
$ a(n) < \frac{{{n^2} \cdot {{(ln(n))}^2}}}{2}$
