I know for a fact that for all $n > 5040$ there is no violation for robin's inequality from square-full,square-free numbers,i also know that in order for a number to violate robin's inequality then the number have the form :
$N =p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r} * p_{r+1} p_{r+2} \cdots p_{m}$
such that $k_i \geq 2$ for all $i$ and the prime in ascending order, also the $k_i$ sequence is non-increasing sequence.
all the above i know from paul erdos paper on the matter,but what i really don't understand is the following
does mathematicans know that $p_r = O(\sqrt{p_m}) $ ?!, i think erdos paper give this bound on $p_r$ but i am not sure, any ref to a paper proving these bounds or deeper insight to the problem would be most appreciated.
Thanks