An idea that seemed to work:
Too small a number of divisors doesn't seem likely.
What if there are 6 divisors? The numbers could be of the form p*q^2.
For one of them p=2 and the other q=2.
These numbers would be 2 apart
So we'd have 2*q^2 = p*2^2 ±2
I then made a short list of potential candidates where both p and q are prime: (p,q)
(5,3)
(13,5)
(61,11)
(181,19)
I stopped here thinking I'd increase my search if needed.
These pairs yield two numbers with 6 factors that are 2 apart:
18 and 20
50 and 52
242 and 244
722 and 724
Next to check if the number between has 6 factors:
19  no (just 2)
51  no (just 4)
243  yes this is 3^5 (I hadn't thought of that)
723  no (just 4)
so we have 3 numbers 242,243,244 and need a 4th at either end.
241 is prime
245 = 5*7^2 does it!
n=242[In hindsight this could have been easier and there were a lot of pitfalls but I looked this up https://oeis.org/A006558 so this answer is right.]

Posted by Jer
on 20140620 11:48:18 