[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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