Consecutive Squarefrees, Not Coprime

[David Wilson]

Sorry, I did send a reply to Jud McCranie and meant to copy to seqfan.

----- Original Message ----- 
From: "Don Reble" <djr at nk.ca>
To: "Seqfan" <seqfan at ext.jussieu.fr>
Sent: Sunday, August 21, 2005 1:14 AM
Subject: Re: Consecutive Squarefrees, Not Coprime


> David, seqfans:
>
>> 6 <=1292013541080148674
>> I'm sure my bound on a(6) is rather loose and could be tightened with
>> a little effort.

Yes, and it looks like Don has applied the effort.

>> Did you mean "what is the smallest consecutive squarefree a, b such that
>> gcd(a, b) = n"? If so, I can confirm 1,2,3,5,7;
>
>    I believe David does mean that.

Indeed.  Alzheimer's is hell, you know.

>> for a(6), did you mean ">=" rather than "<="?
>
>    I believe David meant what he said, "<=".

Yes, I was accurate here.  1292013541080148674 is not a test limit, it is a 
constructed example of consecutive squarefree numbers with gcd 6, and 
therefore is an upper bound on a(6), the smallest such number.

>    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.)

Yes, thank you for your efforts, you have tightened the bound considerably, 
to within the 64-bit range.  We might now hope for a solution.

> -- 
> Don Reble  djr at nk.ca