[seqfan] Re: A New Category of Sequences?

[Fred Lunnon]

  A search for "Legendre conjecture" turns up a large number of recent claims
to have proved it, and cursory sampling suggests that the expected quality
is dismal.  Example: one advertised Ph. D. (subject unspecified) devotes a
substantial proportion of their contribution to the rediscovery of the sieve of
Eratosthenes (274--196 BC).  The tradition of seeking fame and fortune via
a proof of Fermat's "last theorem" has apparently migrated in the direction
of Landau's list of problems, after some spoilsport actually did so ...

  But I feel obliged to issue a plea to anybody inspired to investigate such
deceptively easily-stated classical problems.  Before you expend time and
effort attempting to communicate the results of your research to the world,
do take the trouble at least to establish what is already known about the
matter, as well as what constitutes a rigorous mathematical proof.

  A good start might be to read (and understand) Hardy & Wright ...

WFL



On 7/22/21, Jerry <zhivago47 at gmail.com> wrote:
> In studying the Legendre Conjecture, I have created a set of sequences to
> measure the density (among integers) of primes where n^2 < p < (n+1)^2.
> However, this set is infinite. I have considered  a couple of ways to
> represent this set of sequences in OEIS, but if the question has already
> been addressed, then I would simply use the format that OEIS has decided
> upon. I have not been able to find any such sets of sequences in the OEIS,
> but that is more likely to be because I don't know how to search for them
> than because no such sets of sequences exist in the OEIS. So if anyone can
> direct me to an example that shows how such sets should be represented in
> OEIS, I would very much appreciate it. If it turns out that this type of
> set of sequences is in fact new to OEIS, then I would explain the ways to
> represent it that I have thought of and ask for feedback on those.
>
> Notes
>
> 1. About 7 weeks ago, I submitted to this list a potential sequence for
> OEIS related to Legendre's Conjecture (SeqFan Digest, Vol 153, Issue 2).
> The discussion ended rather quickly, and I realized I needed to give my
> ideas more thought. I have thought through much more thoroughly now what I
> was looking for to study the Conjecture, and I will summarize below what my
> proposed set of sequences is, and how I arrived at it.
>
> 2. My basic goal is to evaluate the density (among integers) of primes
> between consecutive integer squares. The simplest way to measure this
> density is to divide each term in A014085 by the corresponding term in
> A005843 (excluding n = 0 in each sequence). (Note, however, that neither of
> these sequences is cross-referenced in the other. After I have resolved the
> question of this post, I will attempt to edit the two sequences to note the
> fractional sequence they can represent.)
>
> This fractional sequence can also be represented with decimals of course,
> but I found that less precise, since many decimals cannot give exact values
> (e.g., .33333 vs 1/3). I felt that if I could represent the sequence as
> exact integers, I might be able to explore the relationships of the terms
> of the sequence more intuitively. And there is a very easy way to do this,
> namely by calculating the LCM of the fractions, then multiplying all terms
> by that LCM.
>
> Of course, one cannot calculate the LCM of an infinite sequence of
> fractions. Instead, one must choose a finite subset of the sequence and
> multiply each term of that finite sequence by its LCM. If one starts always
> with the first term of the infinite sequence, then there is one finite
> sequence of integers for each member of the infinite sequence,
> specifically, the sequence ending with the term a(n), with n including all
> positive integers.
>
> Properly speaking, each of these finite sequences is a fractional sequence,
> with the denominator for all terms in the sequence being the LCM for that
> sequence. Since the first fraction in the infinite sequence is 1/1, the
> first numerator in each finite sequence provides the LCM, and so the
> denominator for all terms of the finite sequence. Hence, there is no need
> to add a sequence to the OEIS for the denominators of each of the finite
> sequences, which would be  'forbidden' monomial sequences in any case.
>
> The only OEIS entry that would be needed related to the denominators of the
> finite sequences would be the sequence of LCMs for a(n) of the infinite
> sequence.
>
> As a simple example, I show here the finite fractional sequence ending in
> a(10) and the finite integer sequence ending in a(10).
>
> 1, 1/2, 1/3, 3/8, 1/5, 1/3, 3/14, 1/4, 1/6, 1/4
>
>
> 840, 420, 280, 315, 168, 280, 180, 210, 140, 210
>
> Note that the first term in the second sequence above is the LCM of the
> first sequence, and so the denominator of all terms in the second sequence.
>
> I don't want to spam this list with more examples, although I would like to
> make them available to anyone who might want to look at them. However, I do
> not seem to be able to enter any text on my User Page, although I am logged
> in on the OEIS Wiki. If someone can explain to me how to put information on
> my User Page, I will show there the finite integer sequences for a(20),
> a(30), a(40), and a(50), as well as the first 50 terms of the LCM sequence.
>
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