[seqfan] A007843, A007844, A007845: What is definitively known?
Nathan Fox
nhf12 at scarletmail.rutgers.edu
Tue Dec 24 20:45:02 CET 2019
Dear seqfans,
I've stumbled across sequences A007843, A007844, and A007845 (least
positive integer k for which p^k divides k!, for p=2, p=3, and p=5
respectively). These sequences have some properties that are, in some
cases, stated tentatively in comments. Here are my observations:
* The first comment on A007843 indicates that each integer appears its
2-adic valuation (A007814) number of times. A similar statement
about 5-adic valuation is present on A007845. Such a comment is
missing on A007844 regarding 3-adic valuations, but the first
comment speculates that all terms are divisible by 3. It would seem
that such a statement about p-adic valuation is true for all primes p.
* 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.
* At least for A007843, there is a proof for the meta-Fibonacci
connection (given in the first link below the b-file).
* These sequences have a PDF attached from F. Smarandache (1993)
mentioning them in the context of open problems (Problems 47, 48,
and 49 in the PDF). 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.
Can anyone provide a definitive statement on what is known versus what
is only conjectured here? Also, what is the preferred method of
editing/adding comments to such sequences clarifying
speculative-sounding comments?
Thank you,
Nathan
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