[seqfan] Re: Prime numbers that differ by 30 with no primes between p and p+30

Thu Dec 26 12:09:42 CET 2019

> I think that what I find different is that it seems that a number
> of sequences appear to address only one-half of the full sequence.
> I know, each half is a sequence; but is not both halves a sequence
> also? ...

I would look at it this way: the 'first half' is integers n with the property that:

  (n > 0) AND (n is prime) AND (n+30 is prime) AND (n+i is composite for i=1, 2, ... 29)

The 'second half' is integers with the different property that:

  (n > 30) AND (n is prime) AND (n-30 is prime) AND (n-i is composite for i=1, 2, ... 29)

So the two halves put together is integers n with a more complicated property, namely the OR of those two properties. In general, one has to have a limit on how complex a condition satisfied by an OEIS sequence is allowed to be, and even the extra complexity of allowing the OR of two different sequences' conditions would raise the number of OEIS sequences from its current 300000-odd to unmanageable billions...

Put another way, if one expects 4297,4327,4831,4861,5351,5381,5749,5779,6491,6521,... to be an OEIS sequence, shouldn't one also expect 1,2,3,6,9,24,27,81,120,243,720,729,2187,5040,... to be one? It is the sequence of n such that n is a factorial OR n is a power of 3 - i.e. the results of combining A000142 and A000244 - not clearly any more complex than combining A124596 and A124596 + 30.

Basically, I think the issue here is more one about looking up sequences than about storing sequences: the ideal solution would be that if OEIS was asked to look up 4297,4327,4831,4861,5351,5381,5749,5779,6491,6521,..., it would tell you that it matches (A124596) OR (A124596+30), not that 4297,4327,4831,4861,5351,5381,5749,5779,6491,6521,... should actually be an extra OEIS sequence. Unfortunately, that same billions-of-possibilities issue means that it's not very practical to do such lookups routinely...

It *is* possible to do them non-routinely - see what https://oeis.org/ol.html says about Superseeker for how. I have just tried giving Superseeker the input line "lookup 4297 4327 4831 4861 5351 5381 5749 5779 6491 6521" and got the response:

==========

a(n) = 4297, 4327, 4831, 4861, 5351, 5381, 5749, 5779, 6491, 6521

Note: Your sequence does not directly appear in the OEIS.
If it is of general interest, please submit it at https://oeis.org/Submit.html.

# Transformations

These sequences match transformations of the original query.

T006 a(n) for odd n
  = 4297, 4831, 5351, 5749, 6491

oeis.org/A124596 Primes p such that q-p = 30, where q is the next prime 
                          after p.

      <4297, 4831, 5351, 5749, 6491>, 6917, 7253, 7759, 7963, 8389, 8893, 
      13063, 13187, 13933, 13967, 14251, 14983, 16381, 16573, 17627, 18553, 
      18869, 20563, 21283, 21347, 21617, 23633, 23689, 24251, 25189, 26053, 
      26597, 27299, 27367, 27551, 28319, 28979, 29537

In transformation descriptions,
Sn(z) denotes the ordinary generating function with coefficients a(n), and
En(z) denotes the exponential generating function with coefficients a(n).

==========

It's not a full explanation of your sequence - it's failed to pick up on the even terms being 30 more than A124596 - but it's enough to put one on the right track...

I do see some arguments in the opposite direction, by the way - the two conditions "(n > 0) AND (n is prime) AND (n+30 is prime) AND (n+i is composite for i=1, 2, ... 29)" and "(n > 30) AND (n is prime) AND (n-30 is prime) AND (n-i is composite for i=1, 2, ... 29)" are more related to each other than in my example of "n is a power of 3" and "n is a factorial", which might be regarded (but AFAIAA isn't) as a reason to have the additional sequence for it. So I'm not expressing a strong view on whether it should be OEIS policy to include sequences such as 4297,4327,4831,4861,5351,5381,5749,5779,6491,6521,... - I'm just saying that it appears to me that it currently isn't.

One final comment is that if it isn't already there (I haven't checked exhaustively!), there might be something about this sort of situation that could usefully be added to a 'help' or 'hints' file - something along the lines of "if you have a sequence of pairs (or triplets, etc) of numbers and don't find it in the OEIS, try looking up the sequences of first numbers and second numbers in the pairs separately".

David

> On 25 December 2019 at 17:01 Harry Neel <neelh48 at hotmail.com> wrote:
> 
> 
> 
> Thanks,
> 
> I was able to figure out that only the first term was provided because of the definition of the sequence.  It is awkward for me because I was expecting to see both values.  It took some double-checking of the definition  for me to be sure I understood it properly.
> 
> I think that what I find different is that it seems that a number of sequences appear to address only one-half of the full sequence.  I know, each half is a sequence; but is not both halves a sequence also?  I have not searched, but what would I find if I searched for the second term and not the first term of the sequence provided by A124596; and should it be listed in the cross-references?
> 
> Thanks Everyone'
> 
> H. Neel
> 
> Sent from Outlook<http://aka.ms/weboutlook>
> 
> ________________________________
> From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of zak seidov via SeqFan <seqfan at list.seqfan.eu>
> Sent: Wednesday, December 25, 2019 3:34 AM
> To: seqfan at list.seqfan.eu <seqfan at list.seqfan.eu>
> Cc: zak seidov <zakseidov at yahoo.com>
> Subject: [seqfan] Re: Prime numbers that differ by 30 with no primes between p and p+30
> 
> A124596 Primes p such that q-p = 30, where q is the next prime after p.
>  4297, 4831, 5351, 5749, 6491, 6917, 7253
> 
>     On Wednesday, December 25, 2019, 7:28:30 AM GMT+2, Harry Neel <neelh48 at hotmail.com> wrote:
> 
>  While I have been able to obtain a sequence containing only the first prime of the sequence when I enter
> 
>                 4297,4327,4831,4861,5351,5381
> 
> the return is “Sorry….”
> 
> As these are distinct prime numbers that have no primes between them, should I receive a different response.
> 
> Hopefully I am not missing terms.
> 
> Thanks,
> 
> H. Neel
> 
> Sent from Mail<https://go.microsoft.com/fwlink/?LinkId=550986> for Windows 10
> 
> 
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