# [seqfan] Re: Nomination for 250000.

M. F. Hasler seqfan at hasler.fr
Sun Nov 23 15:33:11 CET 2014

```On Sat, Nov 15, 2014 at 7:51 AM, Neil Sloane <njasloane at gmail.com> wrote:
> Dear Juri,
> Please submit these two sequences to the OEIS,
> and tell me the A-numbers:
>
> "Numbers n such that n, 2^n - 1 and binomial coefficient(2^n - 1, n) are
> all squarefree: 1, 2, 3, 11, 29, 31, 51, 55, 57, ... (finite)

https://oeis.org/A245569 : Numbers n such that binomial(2^n-1,n) is squarefree.
0, 1, 2, 3, 4, 6, 11, 12, 21, 28, 29, 31, 51, 54, 55, 57
Motivated by the existence of the subsequence A246699 of squarefree
terms in this sequence.

Actually
- the additional restriction on n only removes 4 terms 4, 12, 28, 54
(for such a short sequence it could be frowned upon adding the
subsequence of squarefree terms and the further subsequence of (5)
primes.... how about subsequences of non-squarefree (in analogy to
A237043), odd and even terms ? [This was not a suggestion...])
- the other seq. (A246699, now has an additional 21 and does not have
any more "binomial" in the NAME, which makes it impossible to find it
based upon the previous mail...) says "fini"(te) without comment or
reference. I think this should be fixed.
- I don't know a reference and/or proof of the finiteness, so I
preferred to add "conjectured to be...", at the risk of exhibiting my
ignorance if this is a (well(?)) konwn result...

Thanks for adding such information if it exists.

Maximilian

> Primes p such that 2^p - 1 is not squarefree: 359, 397, 419, ...
> (infinite). "
>
> By the way, the second one should have a cross-reference
> saying "These are the primes in A049094."
>
> Thank you
>
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Email: njasloane at gmail.com
>
>
> On Fri, Nov 14, 2014 at 5:51 PM, юрий герасимов <2stepan at rambler.ru> wrote:
>
>>
>> Numbers n such that n, 2^n - 1 and binomial coefficient(2^n - 1, n) are
>> all squarefree: 1, 2, 3, 11, 29, 31, 51, 55, 57, ... (finite)
>> or Primes p such that 2^p - 1 is not squarefree: 359, 397, 419, ...
>> (infinite). JSG.
>

```