[seqfan] A New Category of Sequences?
Jerry
zhivago47 at gmail.com
Thu Jul 22 21:58:06 CEST 2021
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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