[seqfan] Sum of squares -- and a concatenation

[Eric Angelini]

Hello Seqfans,

----------------Step #1------------------
We start with an integer A;
We express A as a sum of squares:

A = 113

113 = 100 + 9 + 4 = 10^2 + 3^2 + 4^2   (a)
or
113 = 81 + 16 + 16 = 9^2 + 4^2 + 4^2   (b)
or
113 = 64 + 49 = 8^2 + 7^2              (c)
or
113 = 64 + 49 + 0 = 8^2 + 7^2 + 0^2    (d)
etc.

----------------Step #2------------------
We build the smallest possible integer B,
B is the "minimal concatenation" of the 
  terms squared in step #1:

Tries:
according to (a) we could have B = 1034 or 1043 or 4310, etc.
according to (b) we could have B = 944 or 494 or 449
according to (c) we could have B = 87 or 78
according to (d) we could have B = 708 or 780 or 807 or 870. 
etc.
It seems that the "minimal" B is 78.

----------------Step #3------------------
We iterate.
----------------Example------------------
Starting S with a(1)=2:

S = 2,11,113,78,257,116,104,102,277,...

Explanation:

2 = 1 + 1 = 1^2 + 1^2 --> 11
11 = 1 + 1 + 9 = 1^2 +1^2 +3^2 --> 113
113 = 49 + 64 = 7^2 + 8^2 --> 78
78 = 4 + 25 + 49 = 2^2 + 5^2 + 7^2 --> 257
257 = 1 + 256 = 1^2 + 16^2 --> 116
116 = 100 + 16 = 10^2 + 4^2 --> 104
104 = 100 + 4 = 10^2 + 2^2 --> 102
102 = 4 + 49 + 49 = 2^2 + 7^2 + 7^2 --> 277
...

The problem here is to find at each step "the minimal B" --
thus I cannot guarantee that S is correct so far.

Is this a "universe" worth to explore? (loops, fixed points,
flights...)
Best,
É.