[seqfan] Re: Shortest sequence to get prime

[Neil Fernandez]

Hi again Alonso,

Another sequence we can make is

a(n):= smallest odd prime p for which np is the smallest multiple of p
such that np-1 is square-free

This begins 3, 17, 5, 113, 4019, 6701, 84317, 1284817, 39957487, ...

3     (3*1 - 1    = 2),
17    (17*2 - 1   = 3 * 11),
5     (5*3 - 1    = 2 * 7),
113   (113*4 - 1  = 11 * 41),
4019  (4019*5 - 1 = 2 * 3 * 17 * 97),
6701  (6701*6 - 1 = 5 * 11 * 17 * 43),
84317 (84317*7 - 1       = 2 * 43 * 6863),
1284817 (1294817*8 - 1   = 3 * 5 * 11 * 67 * 937),
39957487 (39957487*9 - 1 = 2 * 223 * 806317),
...


Neil

In message <CAGyGvfU-VOb2qyGXQbYi7PZy3zjAfSSQzi2-1RzBQXj1oCjSEQ at mail.gma
il.com>, Alonso Del Arte <alonso.delarte at gmail.com> writes

>Given a positive prime *p*, what is is the shortest, non-empty finite list
>of the smallest distinct positive primes that does not contain *p* such
>that the product of those distinct primes is one less than a multiple of *p*?
>As you can see from the following list, for some primes I run into a little
>bit of a conundrum.
>
>  3 {2}
>  5 {2, 7}
>  7 {2, 3}
> 11 {2, 5}
> 13 {2, 19} OR {3, 17}?
> 17 {3, 11}
> 19 {37}
> 23 {2, 11}
> 29 {3, 19}
> 31 {2, 3, 5} OR {61}?
> 37 {73}
> 41 {2, 61}
> 43 {2, 3, 7} OR {5, 17}?
> 47 {2, 23}
> 53 {3, 5, 7} OR {2, 79}?
> 59 {2, 29}
> 61 {2, 7, 13}
> 67 {2, 3, 11} OR {7, 19}?
> 71 {2, 5, 7} OR {3, 47}?
> 73 {5, 29}
> 79 {2, 3, 13} OR {157}?
> 83 {2, 41}
> 89 {3, 59}
> 97 {193}