[seqfan] Re: What in the next one? [primes isolated from squarefrees]

[Chris Thompson]

On Jan 26 2016, Neil Sloane wrote:

>Chris, what was the definition of this one? 2, 17, 727, 47527, 29002021
>(note typo above), 494501773?
>
>Are you looking at the squarefree nonprimes S, and  asking
>for the smallest prime p such that the nearest element of S differs from p
>by at least n (starting at n=1)?
>No, that doesn't work. I'm having trouble seeing how you got 2 and then 17
>rather than 5 or 7.

Well, I didn't originate the sequence (that was юрий герасимов, which I
think transliterates as Yuri Gyerasimov) but I took it to mean a(n) to be
the smallest number such that

   a(n) is prime
   a(n)+i and a(n)-i are not squarefree for 1<=1<=n

(If only there were a simple word for "not squarefree"! But "squareful"
is stolen by A001694...) E.g. 17 is prime, 16 and 18 are not squarefree.

>Could you please submit it to the OEIS, and also the other one you mention (1,
>17, 26, 2526, 5876126, 8061827, ...)?

This one being, by contrast, being the smallest number such that

   a(n) is squarefree
   a(n)-i and a(n)+i are not squarefree for 1<=i<=n

Or maybe the series would be better renumbered by using 1<=i<n ("at least
n distant from the nearest squarefree number", instead of "more than n
distant..."). That would match the numbering for the "isolated primes",
A023186.

As regards putting them in OEIS, I'll consider doing that for the second,
but I rather dislike the arbitrary nature of the first one. (Why mix
primes with squarefrees? Why not squarefrees isolated from primes, or
all sorts of other possible mixtures?) Although the easiest sledgehammer
to prove that the second series is infinite is maybe to prove that the
first one is. Choose different p_i^2 to divide each x+i, -n<=i<=n, i!=0,
and also p_i not dividing x. Then CRT gives an AP of values for x in which
Dirichlet's Theorem shows there are primes.

-- 
Chris Thompson
Email: cet1 at cam.ac.uk