complex perfect numbers

[Ed Pegg Jr]

David Wasserman wrote:
 > I couldn't find a standard definition of "perfect" in the Gaussian 
integers, so I'm in favor of using any definition that 1) is somehow 
related to the usual definition in N, and 2) leads to an interesting 
sequence.  Just make sure you specify the definition.

The "Perfect Abs" sequence could be considered

5+3i, 3+7i, 19+8i, 15+42i, 6+57i, 29+48i, 19+82i, 74+33i, 111+78i,
147+77i, 185+83i, 91+189i, 197+154i, 269+92i, 122+321i, 159+341i,
72+549i

These are numbers k where Abs[(Total[Divisors[k]] - k)] = Abs[k]

For example, the divisors for 269+92i are: 1, 2+I, 3+4i, 6+5i, 7+2i,
7+16i, 12+11i, 13+34i, 17+126i, 32+47i, 39+2i, 269+92i.
The (sum - k) is 139+248i.  Abs[139+248i] == Abs[269+92i]

Perfect Abs is a weaker result than Perfect.  But they have the
advantage of existing.

--Ed Pegg Jr