[seqfan] Re: A New Category of Sequences?

[Allan Wechsler]

If I understand Jerry's thoughts from the earlier thread correctly:

Between n^2 and (n+1)^2 exclusive, there are 2n integers.

Of these 2n integers, A014085(n) are prime.

Legendre's conjecture is that A014085(n) > 0 for all n. Jerry, quite
reasonably, considers the rational sequence A014085(n)/2n, the prevalence
of primes between n^2 and (n+1)^2. The prime number theorem suggests that
this prevalence falls very slowly, like 1/log(n).

Jerry thinks some insight might be gained by taking the first k elements of
this rational sequence (for an arbitrarily chosen k) and converting them to
integers by multiplying them all by the least common multiple of their
denominators (in lowest terms).

The problem he posed was, how do you present an infinite sequence of finite
sequences in OEIS? One obvious suggestion is just to concatenate them. The
segments will be completely regular, beginning and ending at triangular
indices (1, 2-3, 4-6, 7-10, and so on). It won't even be very hard to give
a "formula" in terms of A014085.

I personally don't see what insight will be gained from this that isn't
revealed by A014085. The segments will, in general, subside like 1/log(n),
each segment scaled by a larger and larger, increasingly composite factor.
But it wouldn't be the least-well-motivated sequence in the Encyclopedia,
not by a long shot.

If this sequence is added, I would also recommend adding the numerator and
denominator sequences of the underlying rational sequence (in lowest
terms), and a separate sequence for the LCM of the denominators up through
n.

On Tue, Jul 27, 2021 at 1:19 PM Elijah Beregovsky <
elijah.beregovsky at gmail.com> wrote:

> I might be misunderstanding your problem, but as far as I know, as long
> your sequence family is countably infinite and has a meaningful ordering,
> you could enter them as one table read by antidiagonals. Like the
> multifactorial array https://oeis.org/A114423. Does this help? If not,
> please tell more about your sequences
> Elijah Beregovsky
>
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