Recurring Digital Invariant
المؤلف:
Madachy, J. S.
المصدر:
Madachy,s Mathematical Recreations. New York: Dover
الجزء والصفحة:
pp. 163-165
16-11-2020
877
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,
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 |
|
REFERENCES:
Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 163-165, 1979.
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