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

israel at math.ubc.ca israel at math.ubc.ca
Wed Oct 24 21:57:30 CEST 2018

```That there are infinitely many such primes is Polignac's conjecture.
But you should have a(0)=2 and a(1)=4, and then your sequence is A087711.

Cheers,
Robert

On Oct 24 2018, Allan Wechsler wrote:

>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/
>>
>
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>
>

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