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

T. D. Noe noe at sspectra.com
Sat Nov 7 01:52:06 CET 2009

```>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 terms 2 to 33 are
11, 17, 83, 47, 0, 107, 227, 569, 59051, 317, 0, 9479, 2027, 947, 0, 2207,
0, 2837, 88211, 295247, 0, 3467, 50627, 9034499, 11027, 47387, 0, 14177, 0,
15017, 1476227

Changing the offset and untying this sequence from the orderly numbers,
this could be called:

Least prime p such that p-2 has n divisors, or 0 if no prime exists.

3, 5, 11, 17, 83, 47, 0, 107, 227, 569, 59051, 317, 0, 9479, 2027, 947, 0,
2207, 0, 2837, 88211, 295247, 0, 3467, 50627, 9034499, 11027, 47387, 0,
14177, 0, 15017, 1476227

For this sequence, the terms a[p] for prime p are 0 except when 3^(p-1)+2
is prime.  Using A051783, we find the exceptional primes to be p=2, 3, 5,
11, 37, 127, 6959....  For these p, a[p] = 3^(p-1)+2.

Similar to A066814 (Smallest prime p such that p-1 has n divisors, or 0 if
no such prime exists.).

Tony

```