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

[Andrew Weimholt]

Only prime orderly numbers appear to have multiple values of k, so for
the following remarks, let's assume that is the case. The following is
a more generalized version of Tony's remarks about the non-existence
of an orderly number with precisely 6 values of k....

The only numbers with a prime number of divisors are powers of primes.
If q is prime, then q^x has x+1 divisors. So to find an orderly number
with x values of k, where x+1 is prime, we must look for a prime of
the form q^x+2. All primes greater than 3 are congurent to +/- 1 mod
6, so if x is even (which it is if x+1 is an odd prime), then q>3 ==>
q^x is conguent to 1 mod 6, which means q^x + 2 is divisible by 3 and
therefore not prime.
Therefore the only primes of the form q^x + 2, where x>0 is even, are
those with q=3.
So the only orderly number with precisely 2 values of k is 11, because
11 = 3^2 + 2
The only orderly number with precisely 4 values of k is 83, because 83 = 3^4 + 2
3^6 + 2 is NOT prime, so there is no orderly number with precisely 6
values of k.
3^10 + 2 is prime (59051), so it is the only orderly number with
precisely 10 values of k
3^12 + 2 is NOT prime, so there is no orderly number with preciseuly
12 values of k.

For precisely 13 values of k, since 13+1 is not prime, you can also
look for primes of the form q*r^6+2 where q and r are prime.
You can try
3*r^6 + 2
q*3^6 + 2
or
q*r^6 + 2, where q == -1 mod 6

Andrew

On 11/6/09, Alonso Del Arte <alonso.delarte at gmail.com> wrote:
> 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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