# [seqfan] Re: finite a(n) for q=0 and infinite a(n) for q>0?

Allan Wechsler acwacw at gmail.com
Wed Oct 24 20:04:51 CEST 2018

```It is always my habit to look to see if all the simpler concepts than a
given proposal are already in the Encyclopedia. In this case, I have just
discovered that we seem to be missing "a(n) = smallest k such that k +/- n
are both prime"!

a(1) = 2 (2,2)
a(2) = 5 (3,7)
a(3) = 8 (5,11)
a(4) = 7 (3,11)
a(5) = 8 (3,13)
a(6) = 11 (5,17)

(2,5,8,7,8,11 ...) is already not in the Encyclopedia.

I suppose it is an article of faith that for all n, there are always two
primes separated by 2n. This is a Goldbach-like conjecture -- I see nothing
forcing it to be true, but I would happily bet any amount that it is. The
related sequence, "a(n) = smallest prime p such that p+2n is also prime",
which I have written in the table above as the first of the parenthesized
prime pair, is also apparently not in OEIS.

None of this is to disparage Yuri Gerasimov's work -- it was just a simpler
idea suggested to me by his sequences.

On Wed, Oct 24, 2018 at 1:25 PM M. F. Hasler <seqfan at hasler.fr> wrote:

> I don't understand the corrected version.
> (Euphemism. The "m = n ..." is problematic.)
> Is a_q(n) no more the smallest k such that
> #{ d>q : d|k and d±q both are prime} = n
> ?
>
> --
> Maximilian
>
>
> On Wed, 24 Oct 2018, 09:37 юрий герасимов <2stepan at rambler.ru> wrote:
>
> > Dear Allan and Maximilian, Thank you. I agree with your reasoning.
> > Sequence a(n)
> > for q=0 is A002110(n). Correction of a(n) for q>0: a(n) is the smallest k
> > such
> > that number of some divisors m of k is equal to excatly n where m = n <
> > A000005(k) and m-q, m+q are both prime. JSG.
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```