[IntSeq] Friends of R and friends of S

[Rainer Rosenthal]

Rainer Rosenthal wrote

> R_2 = 0,1,2,3,4,5,6,...      = "the natural numbers"
> R_3 = 0,1,3,8,21,55,144,...  = A001906
> R_6 = 0,1,6,35,204,1189,...  = A001109
>
> S_4 = 0,1,4,9,16,25,36,49,...  = A000290
> S_9 = 0,1,9,64,441,3025,20736,...  = A049684
> S_36 = 0,1,36,1225,41616,1413721,... = A001110(x)

Well, it's no surprise (understatement, hehe), to
find S_{r^2} = {R_r}^2, as can be seen from
S_4 = {R_2}^2, S_9 = {R_3}^2 and S_36 = {R_6}^2.

Other examples are:

> R_7 = A004187
> R_12 = A004191
> S_49 = 0,1,49,2304,108241, ...     = A049682
> S_144 = 0,1,12^2,143^2,1704^2,...  = A004191^2

This gives a lot of interrelations between many sequences
already in the OEIS(*) of Neil Sloane! I am happy to search
the archive accordingly to provide the appropriate
comments (hopefully this weekend).

The best thing is the very very elementary character of
these interrelations:

   ~~~~~~~~~~~~~~~~~~~~~~~~~~
    If A and B are R-friends
       then A^2 and B^2
         are S-friends.
   ~~~~~~~~~~~~~~~~~~~~~~~~~~

Proof: Let A and B be R-friends,
       i.e. A | (B^2-1) and B | (A^2-1).
       Then A^2 | (B^2-1)^2 and B^2 | (A^2-1)^2,
       i.e. A^2 and B^2 are S-friends.

(*) for readers of alt.math.recreational: The OEIS is found at
http://www.research.att.com/~njas/sequences/

Best regards, especially to Gottfried Helms,
Rainer Rosenthal
r.rosenthal at web.de