A072507

[Dean Hickerson]

Mostly to Franklin T. Adams-Watters:

> The next question is, what is the length, for a given n, of the longest 
> sequence of integers with exactly n divisors?
...
> 10 2 to 7

The maximum is 3, as in this example:

    7939375 = 5^4 12703
    7939376 = 2^4 496211
    7939377 = 3^4 98017

If there were 4, then 2 of them, say n and n+2, must be even, and hence equal
to either  2^9,  2^4 p,  or  2 p^4  for some odd prime p.  The first doesn't
work, since neither 510 nor 514 has the right form.  Since one of the 2
numbers is divisible by 4 and the other is not, we either have

    n = 2 p^4  and  n+2 = 2^4 q.

or

    n = 2^4 p  and  n+2 = 2 q^4

The first case implies that  p^4 == 7 (mod 8)  and the second implies (since
p is odd) that  q^4 == 9 (mod 16).  Both of these are impossible.

Dean Hickerson
dean at math.ucdavis.edu