Consecutive Squarefrees, Not Coprime

[Jud McCranie]

Leroy Quet proposed this.  David Wilson replied:

>The gcd of any two consecutive squarefree numbers must be 
>squarefree.  Given the k-tuple conjecture, I can show that every 
>squarefree number is in fact the gcd of two consecutive squarefrees (the 
>k-tuple conjecture may not be necessary here).
>
>This leads us immediately to ask, for squarefree n, what is the smallest a 
>such that a and a+n are consecutive squarefrees with gcd(a, a+n) = n.  I get
>
>n a(n)
>1 1
>2 422
>3 174
>5 22830
>6 <=1292013541080148674
>7 234374
>
>I'm sure my bound on a(6) is rather loose and could be tightened with a 
>little effort.  It is not out of the question that the exact value could 
>be found.


Don Reble said:

"Continuing my speculation on David's intentions and methods, the
    sequence
        11 (2) 5 29 31 (6) 23 7 17 (2) 19
    yields a(6) <= 57879439071222. (Yay, the endpoints are squarefree.) "

----------

I've found that with high probability, a(6) = 9216772051242   (and its 
partner is  9216772051254).  At this point I can't be sure that there isn't 
a smaller one, but there probably isn't a smaller one.  Soon I hope to have 
either a proof that it is the smallest (very likely it is) or possibly find 
a smaller one.