The picture-perfect numbers are an intriguing species of perfect numbers introduced by J. L. Pe while investigating
generalized perfect numbers. A (natural) number n is called
picture-perfect (or pp) if its reversal equals the sum of the reversals of the proper divisors of n. Hence, if n is written on one side of an equal sign and its proper divisors on the other, then the resulting equation read backwards is valid--which explains the term "picture-perfect", since a picture of an object is a mirror-image of that object.
Although the first pp number is 6, the first nontrivial pp number is 10311, as can be verified by reading the following equation
backwards:
10311 = 1 + 3 + 7 + 21 + 491 + 1473 + 3437.
(The proper divisors of 10311 appear on the right side.) That is,
7343 + 3741 + 194 + 12 + 7 + 3 + 1 = 11301
is valid.
The pp numbers are extremely rare pearls in the infinite ocean of numbers. The only other pp number less that 10^9 (1 billion) is 21661371.
For more info as well as Mathematica source code to search for these numbers, see
The Picture-Perfect Numbers.
As of the time of writing, a coordinated effort to search for more pp numbers is ongoing. The SUPER (Systematic Undertaking to find Picture-Perfect Numbers) team has currently three members: Daniel Dockery, Mark Ganson, and Joseph Pe. Recently, we completed the search for pp numbers in the range 10^8 to 10^9 without turning up any new ones. We are now investigating the range 10^9 to 10^10.
If you have CPU cycles to spare and access to software such as Mathematica, we invite you to join us in the tantalizing search for these elusive numbers. (To join, please send an email so that we can divide the work according to machine speed.)
Please feel free to post your comments and findings on this discussion forum.