[seqfan] Re: A135044

[Andrew Weimholt]

On Mon, Jan 27, 2014 at 10:13 AM, Charles Greathouse <
charles.greathouse at case.edu> wrote:
> I wrote a quick independent program and also found a(8) = 23.

Charles and Hans,

I don't think your reverse engineering captures the intent of the author.
The title is poorly worded and according to my reverse-engineered
understanding
of the author's intent, just plain 'wrong'.
The ill-conceived terms primeth and composith in the comment only lead to
more confusion, but if you look at the
N. Fernandez paper (which unfortunately uses the term "primeth"), then the
intent becomes a little more clear.

Fernandez is using a sieve like process to arrange the primes in a square
array...

Primes organized in an array according to sieving process described in the
N.Fernandez paper...

        Row[1](n) = 2, 7, 13, 19, 23, ...
        Row[2](n) = 3, 17, 41, 67, 83, ...
        Row[3](n) = 5, 59, 179, ...
        Row[4](n) = 11, 277, ...

        Lets call this  T_p (n, k)

Composites organized in a similar manner, except we use "composite"
numbered positions in our sieve...

        Row[1](n) = 4, 6, 8, 10, 14, 20, 22, ...
        Row[2](n) = 9, 12, 15, 18, 24, ...
        Row[3](n) = 16, 21, 25, ...

        Lets call this T_c (n, k)

If you now take the natural numbers and swap each number (except for one),
with the number
which holds the same spot in the other array, then you get this sequence.

1, 4, 9, 2, 16, 7, 6, 13,
with a(8) = 13
(13 holds the same position in the 'prime' table as 8 does in the
'composite' table).

The best way to fix this sequence is to fix the definition...

Add the T_p(n,k) and T_c(n,k) as two new sequences if not already in OEIS,
(say Axxxxxx and Ayyyyyy
respectively), then the definition of A135044 becomes...

a(1) = 1, a( Axxxxxx(m,k) ) = Ayyyyyy(m,k), and a( Ayyyyyy(m,k) ) =
Axxxxxx(m,k)

The terms primeth and composith in the comment should also be replaced with
better terminology.

Andrew