[seqfan] Methodically filling successive squares and primes with N

[Eric Angelini]

Hello seqFans,

We want the successive squares to be replaced by sums in this way:

 n = 1  2  3   4   5   6   7   8   9   10   11   12 ...
n^2= 1  4  9  16  25  36  49  64  81  100  121  144
     -  -  -  --  --  --  --  --  --  ---  ---  ---
Sum  1  4  2   3   6  11  14  16  18   22   26   30
           +   +   +   +   +   +   +    +    +    +
           7   5   9  12  15  17  19   24   27   32
               +   +   +   +   +   +    +    +    +
               8  10  13  20  31  21   25   28   33
                                   +    +    +    +
                                  23   29   40   49

Thus, to complete a sum, we always use the smallest available positive
integer not used so far and not leading to a contradiction.

The sequence arising from this could be (this is a permutation of N):

Sq = 1,4,2,7,3,5,8,6,9,10,11,12,13,14,15,20,16,17,31,18,19,21,23,...


The same algorithm can be used to "fill the Primes":

  2   3   5   7   11   13   17   19   23   29   31   ...
  -   -   -   -   --   --   --   --   --   --   --
  2   3   1   7    5   13    8   19   11   14   10
          +        +         +         +    +    +
          4        6         9        12   15   21

... producing:

Pr = 2,3,1,4,7,5,6,13,8,9,19,11,12,14,15,10,21,...

Neither "Sq" nor "Pr" are in the OEIS. If someone could check, extend
those seq and submit them (in september -- if they are of interest), 
it would be nice.

Best,
É.