Kaprekar Quintuples

I've written about Kaprekar before in posts titled Kaprekar's Routine on 16th July 2021 and Birth Year Magic on 3rd June 2020. Today, the number associated with my diurnal age (27100) brought me into contact with him again via his Kaprekar Quintuples. These are numbers \(n\) with the property that the digits of \(n^5\) taken \(d\) at a time sum to \(n\) (where \(d\) is the number of digits in \(n\)).

Let's illustrate this in the case of 27100 where we take groups of five digits beginning from the rightmost digit, leaving three digits remaining on the left:$$27100^5 = 14616603103510000000000\\ 146+16603+10351+00000+00000 = 27100 $$Let's try with another number, 7776, that has four digits instead of five. Here we take groups of four digits beginning at the rightmost digit with no digits left over:$$7776^5 = 28430288029929701376 \\2843+0288+0299+2970+1376 = 7776$$The members of the sequence are (permalink showing members up to 100,000):

1, 10, 1000, 7776, 27100, 73440, 95120, 500499, 505791, 540539, 598697, 665335, 697598, 732347, 7607610, 37944478, 46945205, 54995500, 55216205, 56607166, 58106906, 63136413, 66595563, 68167738, 68807564, 69188525, 70667477, 72197730, 73197730, 74145807

These numbers constitute OEIS A171500. These numbers can also be called order-5 Kaprekar numbers and this site shows the initial members. See Table 1.

Table 1

The same site also shows order-4 Kaprekar numbers or Kaprekar quadruples where numbers are raised to the fourth power. See Table 2.

Table 2

Order-3 and order-2 Kaprekar are also shown on this site and could be referred to as Kaprekar triplets/triples and Kaprekar couplets/pairs.