[seqfan] Re: A New Category of Sequences?

[Fred Lunnon]

  Agreed.  One of the most precious qualities of sites like  seqfan  which
still survive in quiet corners of the internet is that a genuine amateur
can raise a completely original question, attempt (however tentatively) to
answer it, in the process acquire new skills, make contact with experts
who may be able to suggest relevant existing tools, and --- just possibly
--- finally unearth a significant discovery.  I would hate to think that such
an explorer had been discouraged by anything I said.

  But, as my brief search emphasised, fools rush in where angels fear to
tread.  There are plenty of the former, and they can quickly submerge
in a tide of garbage anything worthwhile that an actual explorer had to say.
One danger sign is when the question has been well-known for centuries,
and can be parsed using only elementary and commonplace mathematics;
yet the enquirer makes no reference to existing sources.  Beware!

WFL


On 7/23/21, Antti Karttunen <antti.karttunen at gmail.com> wrote:
> On 7/23/21, Fred Lunnon <fred.lunnon at gmail.com> wrote:
>>   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.
>
> Please note that the first poster (Jerry) didn't write anything about
> any proof or an attempt of such, but only about adding some empirical
> data about the topic. But of course for many open problems, such
> empirical data from just a finite b-file range 1 .. 10^(4..5) might be
> highly misleading, even though it could be interesting for other
> reasons. And in any case, such sequences could comment on, or link to
> the resources that show at which point an apparent trend in the data
> holds no more.
>
>
> Best regards,
>
> Antti
>
>
>>
>>   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.
>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>
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>
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