Recurring Digital Invariant
To define a recurring digital invariant of order 
, compute the sum
of the 
th powers of the digits of a number 
. If this number 
is equal to
the original number 
, then 
is called
a 
-Narcissistic
number. If not, compute the sums of the 
th powers of the
digits of 
, and so on. If this process eventually
leads back to the original number 
, the smallest
number in the sequence 
is said to be a 
-recurring digital
invariant. For example,
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
so 55 is an order 3 recurring digital invariant. The following table gives recurring digital invariants of orders 2 to 10 (Madachy 1979).
| order | RDI | cycle lengths |
| 2 | 4 | 8 |
| 3 | 55, 136, 160, 919 | 3, 2, 3, 2 |
| 4 | 1138, 2178 | 7, 2 |
| 5 | 244, 8294, 8299, 9044, 9045, 10933, | 28, 10, 6, 10, 22, 4, 12, 2, 2 |
| 24584, 58618, 89883 | ||
| 6 | 17148, 63804, 93531, 239459, 282595 | 30, 2, 4, 10, 3 |
| 7 | 80441, 86874, 253074, 376762, | 92, 56, 27, 30, 14, 21 |
| 922428, 982108, five more | ||
| 8 | 6822, 7973187, 8616804 | |
| 9 | 322219, 2274831, 20700388, eleven more | |
| 10 | 20818070, five more |
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1+2+3+...+10
