questionable comment in A141399

[Max Alekseyev]

SeqFans,

The comment about finiteness of A141399 does not sound valid to me. It talks about growth of functions omega(x) and gpf(x) but neither of them is monotonic, and so the notion of growth is not defined for them, even asymptotically.

If I do not miss something here, I'd suggest removing the comment and restoring the open state for finiteness and completeness of the sequence.

Please double check.

Regards,

Max

[Sean A. Irvine]

Hi Max,

I have removed the claim. Even if it is true that the sequence is finite, I do not believe that the claimed explanation is sufficient to establish it.

It would seem a finiteness proof would need to rely on the absence of consecutive m-smooth numbers.

Sean.

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[Ruud H.G. van Tol]

Some trivial observations:

A141399 refers to A073491, but A066311 looks more accurate.

Clicking through to A073491, I stumbled on the overloaded meaning of
"prime gap" in the OEIS.
The difference between the value of adjacent prime numbers, and how
consecutive they are, and maybe more.

Examples:
A001223 - Prime gaps: differences between consecutive primes.
A073491 - Numbers having no prime gaps in their factorization.

https://oeis.org/wiki/Index_to_OEIS:_Section_Pri#gaps

I expect that the "secondary meaning" is used much less, but haven't
done the statistics.

-- Ruud

[Sean A. Irvine]

Ken Clements offers more detail in A392163.

https://oeis.org/A392163

If we do not know for certain that A141399 is finite, then it would seem A392163 should not be finite either.

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[Ruud H.G. van Tol]

Somehow that makes me wonder if we already have an April 1 tradition.

Introducing composite gaps wouldn't cut it, sorry.
A sequence of how many proposed cases got refused, with n the month
since OEIS-epoch?
Oh well, just check:
https://oeis.org/search?q=%22april%201%22&sort=number

-- Ruud

[Tomasz Ordowski]

Hello! 

Note that 2*A141399(n) = A194099(n). 

See https://oeis.org/A194099 (fini, full). 

What do you think about this? 

Best, 

Tom Ordo 

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[Ruud H.G. van Tol]

On 2026-03-29 10:42, Tomasz Ordowski wrote:
> Note that 2*A141399(n) = A194099(n).

I see that you mean: A194099(n) = 2 * A141399(n) + 1.
That is also mentioned in the comments of A194099.

> See https://oeis.org/A194099 (fini, full).
> What do you think about this?

Ken Clements recently reasoned that A194099 should be finite, but the
finite-ness of A141399 is mentioned in the reasoning.

-- Ruud

[Tomasz Ordowski]

Yes, my mistake, it should be:

2*A141399(n)+1 = A194099(n).

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[Max Alekseyev]

Hi Tomasz,

I see the same issue in https://oeis.org/A194099 where the comment says:

"the growth rate of the greatest prime divisor (gpf) of m^2-1 exceeds the growth rate of the number of distinct prime divisors (Omega) of m^2-1"

while the growth rate is not well defined, and overall comment sounds like hand waving.

So, I'd suggest removing (fini, full) from this sequence as well.

 

Regards,

Max

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[M F Hasler]

There is a problem in the definition of https://oeis.org/A141399,

"... all the distinct primes that divide k or k+1 are members of a set of consecutive primes."

One can easily guess what is meant, but as it stands, any number satisfies this,

since any prime is a member of a set of consecutive primes. 

(e.g., { p, nextprime(p) }  and even if one would add "...of the same / a common set ...",

one could use (a sufficiently large initial segment of) the set of *all* primes.

Or one could just fill the gaps to get a set of consecutive primes they are all members of.)

What is meant (if I'm not wrong) is that the set of primes that divide k or k+1 *form* a set of consecutive primes,

not just be member of it.

I propose an edit in that sense.

- Maximilian