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

Allan Wechsler acwacw at gmail.com
Wed Oct 24 22:41:05 CEST 2018

```Thank you, Hans and Robert. I did indeed foul up my query, largely due to
getting confused with the first couple of index numbers.

Relevant to Yuri's original suggestion, we have
A000040 for q=0 (mean of prime pairs separated by 0 -- so to speak!)
A014574 for q=1 (mean of prime pairs separated by 2)
A029518 for q=2 (mean of prime pairs separated by 4)
A087695 for q=3 (mean of prime pairs separated by 6)
A087680 for q=4 (mean of prime pairs separated by 8)
A087696 for q=5 (mean of prime pairs separated by 10)

So Yuri's sequences are, for q=0, "a(n) = smallest integer with exactly n
divisors in A000040", for q=1, "a(n) = smallest integer with exactly n
divisors in A014574", and so on. In each case the referenced sequence lists
the "permitted" divisors. Yuri's idea generalizes to almost any of our
positive list sequences, so one could imagine "a(n) = smallest integer with
exactly n Fibonacci divisors", and so on.

On Wed, Oct 24, 2018 at 4:39 PM Allan Wechsler <acwacw at gmail.com> wrote:

> Thank you, Hans and Robert. I did indeed foul up my query, largely due to
> getting confused with the first couple of index numbers.
>
> Relevant to Yuri's original suggestion, we have
> A000040 for q=0 (mean of prime pairs separated by 0 -- so to speak!)
> A014574 for q=1 (mean of prime pairs separated by 2)
> A029518 for q=2 (mean of prime pairs separated by 4)
> A087695 for q=3 (mean of prime pairs separated by 6)
> A087680 for q=4 (mean of prime pairs separated by 8)
> A087696 for q=5 (mean of prime pairs separated by 10)
>
> So Yuri's sequences are, for q=0, "a(n) = smallest integer with exactly n
> divisors in A000040", for q=1, "a(n) = smallest integer with exactly n
> divisors in A014574", and so on. In each case the referenced sequence lists
> the "permitted" divisors. Yuri's idea generalizes to almost any of our
> positive list sequences, so one could imagine "a(n) = smallest integer with
> exactly n Fibonacci divisors", and so on.
>
> I believe
>
> So
>
> On Wed, Oct 24, 2018 at 3:57 PM <israel at math.ubc.ca> wrote:
>
>> 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.
>> >> >
>> >> > --
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>> >> >
>> >>
>> >> --
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>> >>
>> >
>> >--
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>> >
>> >
>>
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>>
>

```