The two groups of integers, 1, 6, 7, 17, 18, 23 and 2, 3, 11, 13, 21, 22 have interesting properties:

Besides the above 12 numbers, there are infinitely many groups of numbers having the above special property.

Let m be any integer and n = 1, 2, 3, 4 or 5. The following identity holds:

The above example is obtained by taking m = 12.

Another identity also holds for n = 1, 2, 3, 4 or 5 and integers a, b and c.

The above example is obtained by taking a = b = 1 and c = 2.

You may try substituting different values of m, a, b and c to obtain your own set of interesting numbers!

If an integer is equal to the sum of nth power each of its digits (when written in decimal), it is called a perfect digital invariant of the nth power.

For example,

Thus 153, 370, 371 and 407 are perfect digital invariant of power 3.