We show that the iteration
$$    A_{n+1} \; = \; \begin{cases}  A_{n}/3 \qquad \; \; \;
 \text{if}  A_{n} \equiv 0 \pmod 3 \\
 [\sum a_{i}]^2 \qquad \text{if} A_{n} \not\equiv 0 \pmod 3
     \end{cases} $$ 
converges for every positive integer $A_{0}$,
  and that for $A_{0} \, = \, 3^{i} \, B$,
        ( $ B  \not\equiv 0 \pmod 3 $ )\\
       $ A_{n}$  converges to the fixed point 1 
       when $ B \equiv 1\; \text{or} \; 8  \pmod 9$ 
       and, \\
      $A_{n}$  converges to the cycle 169 $\longleftrightarrow$ 256
        when $ B \equiv 2 \;\text{or}\; 4 \;\text{or}\;  5\;
           \text{or}\;  7 \; \pmod 9$. \\
        Further, this convergence takes $ i \, + \, O(\, \log^{*} B \,)$ steps.