[seqfan] Re: Shortest sequence to get prime

[M. F. Hasler]

On Mon, Jul 22, 2019 at 11:12 AM Neil Fernandez <> wrote:

> 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, ...


Interesting.
Closer to the initial problem would be:
Least integer m >= 1 such that m*prime(n) - 1 is squarefree.

(This m*p - 1 is the product of the primes in your set of distinct primes
- so this is an "equivalent" encoding of the list you [Alonso] proposed,
which does not require a table (with rows for sets of primes).
A table would require, for unambiguous greedy "decoding", either the
sequence of row lengths, or an unambiguous "marker" for the end of each row.
[Indeed, the sets of primes for p=17 and p=19 show that the last element of
each row is not always (as one might have hoped) larger than the first
element of the next row,
which would have made the sequence-to-table-conversion easier.])

- Maximilian

Alonso Del Arte wrote :
> 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
>


> 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}
> ...
> > 79 {2, 3, 13} OR {157}?
> > 83 {2, 41}
> > 89 {3, 59}
> > 97 {193}
>