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

[T. D. Noe]

>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