[seqfan] Re: A007843, A007844, A007845: What is definitively known?

[M. F. Hasler]

On Tue, Dec 24, 2019 at 9:18 PM Nathan Fox wrote:

> A007843, A007844, and A007845 (least positive integer k for which p^k
> divides k!, for p=2, p=3, and p=5 respectively).


>   * The first comment on A007843 indicates that each integer appears its
>     2-adic valuation (A007814) number of times.(...) It would seem
>     that such a statement about p-adic valuation is true for all primes p.
>

Yes -- as a direct consequence of the following two:

  * There are comments connecting A007843 and A007844 to meta-Fibonacci
>     sequences (A046699 and A120503). The connection between p-adic
>     valuations and term frequencies is definitely known for
> those sequences,

   which would allow dropping the "it appears" qualifiers on these comments.
>

I agree!
Just "be bold" (as they say on Wikipedia) and propose an edit in that sense.
If necessary, one of the editors will improve and/or ask you to improve the
proposal.

  * At least for A007843, there is a proof for the meta-Fibonacci
>     connection (given in the first link below the b-file).
>

This should be mentioned clearly!


>   * These sequences have a PDF attached from F. Smarandache (1993)
>     mentioning them in the context of open problems (...).

It's not clear to me from reading this attachment whether the open problems

are the frequency counts or just further study of these sequences.
>

This is quite general in this an other works from this author.
Let's see it in a positive / constructive way, i.e., second choice:
encouraging further study.


> Can anyone provide a definitive statement on what is known versus what is
> only conjectured here?


Your "definitely known" above settles most of this question, together with
the quoted source with a proof.

Also, what is the preferred method of editing/adding comments to such
> sequences clarifying speculative-sounding comments?
>

A conservative choice is to add (= propose! you can't break anything)
a subsequent comment saying
"The above conjecture / observation by X.Y. [add year or full date added if
incase  ambiguity]
is indeed correct:"
" To see this ...(sketch of proof here)...."
or
"a proof can be found in XXX (2005)." <- reference to LINK/REFERENCES

You may also just add
" [This is true, for a proof see XXX (2005), p.234. - ~~~~] "
or similar, right after the comment, on the same line.

PS: These suggestions are mine, not official OEIS policy. [I'm not aware if
such exists.]
I think/hope most other editors will agree with most of this. Mileage may
vary depending on many factors.

- Maximilian