[seqfan] Re: Orderly Numbers (and related sequences) submitted

Alonso Del Arte alonso.delarte at gmail.com
Fri Nov 6 23:01:11 CET 2009

That makes sense. Something tells me I should be able to extend that same
logic for 6 to the cases of 10, 12 and 13 with relative ease, but I can't,
as if a cloud was blocking the obvious answer from my sight. (The stress
of getting the word out about my Symphony on eBay is getting to me). Despite
the cloud, a sequence of smallest orderly numbers with precisely n values of
k is starting to emerge:

11, 17, 83, 47, [nonexistent], 107, 227, 569, ?, 317, ?, ?, 2027, 947, ...

The subsequence 107, 227, 569 yields a tantalizing result in A117877.

Also, are all these numbers primes? And is 11 the only orderly number with
two values of k?

On Fri, Nov 6, 2009 at 2:21 AM, T. D. Noe <noe at sspectra.com> wrote:

> >For what it's worth, I think A167408 ought to have the keyword "nice." It
> is
> >amazing that for all the years the OEIS has existed, no one had thought of
> >sending this sequence in before. Also, the more I think about the concept,
> >the more tantalizing questions that arise. For instance, what is the
> >smallest orderly number with precisely six values of k? (My calculations
> >suggest it's either a very large number or does not exist).
> I agree that it should be labelled "nice".
> The smallest orderly number with 6 values of k is a prime p such that p-2
> has 6 divisors greater than 1.  This means p-2 is the q^6 for some prime q.
> So just look for the first prime of the form q^6+2.  However, there is no
> such prime because for prime q>3, q^6+2 is divisible by 3.
> BTW, I recently added a comment to the sequence showing that there are an
> infinite number of orderly numbers that are powers of primes.
> Tony
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