[seqfan] Re: More terms for A000445?

[Chris Thompson]

On Dec 18 2019, Brendan McKay wrote:

>A less obscure description would help too.  I found it hard to
>understand even with help from the example.  How is "every large prime"
>in the example relevant to "any prime" in the definition?

Well, the example predates the recent change of definition. The previous
definition was rather vague/misleading, and one could even have read the
example as trying to make sense of it.

The situation for this (and similar sequences, e.g. A330274) that:

1. There is a finite set of small primes for which the pattern doesn't
   occur at all.
2. For all other primes the pattern does occur, and there is an upper
   limit on its first occurrence.
3. There are an infinite number of primes for which that limit is
   attained.

I tried to keep the new definition explanatory without becoming
intolerably verbose. If you think you can do better, please have a go!

Of course there are also variants in which there are is no bound to the
first occurrence of the pattern. The equivalent of A000445 for a sequence
of three consecutive k'th power residues would go

  a(2) = infinite  # see https://doi.org/10.1090/S0002-9939-1962-0138582-6
  a(3) = 23532     # see https://doi.org/10.1090/S0025-5718-1962-0162379-2
  a(4) = infinite  # a(2) infinite => a(k) infinite for any even k
  a(5) unknown, apparently

That's probably not enough terms for an OEIS entry!

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