The prime 127 has 127-1=126 with distinct prime factors 2,3,7 and 127+1=128 with distinct prime factors of only 2; hence 2*3*7=42<127. Log 127/42=q=1.296. Are such primes common? Can a value of q be larger for other primes? I tried before but was not logged in. More examples are 31, with distinct prime factors of 30=2*3*5 and for 32 just 2; hence 2*3*5<31. For 97, 96 has prime factors 2 and 3 and 98 has prime factors 2 and 7; hence 2*3*7=42<97.
[EDIT suggestion to OP: I think the question is asking this: given a prime $p$, calculate the squarefree kernel, $s$, of $p^2-1$ (that is, the product of the distinct prime factors of $p^2-1$); is it common to have $s<p$? is it possible to have $p/s>127/42$? By "common", we may understand "does it happen infinitely often?"] Gerry Myerson understood the question. I can't see why it was closed. Could it be that no one is curious enough to write a program to find more such primes?