[seqfan] Re: A260119; Least positive integer k such that 2^n-1 and k^n-1 are relatively prime.

David Corneth davidacorneth at gmail.com
Thu Sep 3 09:54:59 CEST 2015


Hi Bob,

It's because I conjecture that a(n) is divisible by this product of primes
pp. That means that a(n) is of the form k = pp * m. Then I bluntly use
inspection and test odd m; m = 1, m = 3, m = 5, ... until I find a value k
such that gcd(k^n - 1, 2^n - 1) = 1. I put it up as conjecture because some
k aren't checked at all, it might be that k isn't divisible by pp. But if
it's divisible by pp then this is the least m, which makes the conjectured
values an upperbound. The pattern indeed is irregular and maybe a new
sequence must be created mentioning this m so that it's easier to
investigate them.

For some n, this takes a while to find, for example for n = 9900, m = 8795.

Does that clear things up?
Best,
David

On Thu, Sep 3, 2015 at 9:47 AM, David Corneth <davidacorneth at gmail.com>
wrote:

> Hi Bob,
>
> It's because I conjecture that a(n) is divisible by this product of primes
> pp. That means that a(n) is of the form k = pp * m. Then I bluntly use
> inspection and test m = 1, m = 2, m = 3, ... until I find a value k such
> that gcd(k^n - 1, 2^n - 1) = 1. I put it up as conjecture because some k
> aren't checked at all, it might be that k isn't divisible by pp. But if
> it's divisible by pp then this is the least m, which makes the conjectured
> values an upperbound. The pattern indeed is irregular and maybe a new
> sequence must be created mentioning this m so that it's easier to
> investigate them.
>
> For some n, this takes a while to find, for example for n = 9900, m =
> 8795.
>
> Does that clear things up?
> Best,
> David
>
> On Wed, Sep 2, 2015 at 8:57 AM, Bob Selcoe <rselcoe at entouchonline.net>
> wrote:
>
>> Hi David,
>>
>> I'm a bit confused about your conjectured list up to n=5000.  Why, for
>> example, do you conjecture that m=9 when n=72, m=1 when n=126, and m=5 when
>> n=128?  Why are these upperbounds?
>>
>> Any idea how to calculate m?  The pattern seems quite irregular.
>>
>> Best Wishes,
>> Bob Selcoe
>>
>> --------------------------------------------------
>> From: "David Corneth" <davidacorneth at gmail.com>
>> Sent: Tuesday, September 01, 2015 2:13 PM
>> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
>> Subject: [seqfan] A260119; Least positive integer k such that 2^n-1 and
>> k^n-1 are relatively prime.
>>
>> So I've put a conjecture that might help finding such values of k, but I
>>> don't see a proof and I guess a faster program could be found. There are
>>> some examples to it, but I don't know what characteristic these n have.
>>> The
>>> conjecture is in the history of the sequence, see
>>> https://oeis.org/history/view?seq=A260119&v=36
>>>
>>> Any ideas on how to proceed?
>>>
>>> _______________________________________________
>>>
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>


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