Recurring Digital Invariant
To define a recurring digital invariant of order 
, compute the sum of the th powers of the digits of a number 
. Ifthis number 
is equal to the original number , then 
is called a -Narcissistic Number. If not, computethe 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. Forexample,
 |  |  | |
 | |  | |
 | |  |
so 55 is an order 3 recurring digital invariant. The following table gives recurring digital invariants of orders 2 to 10 (Madachy1979).
| 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.
- Poncelet-Steiner Theorem
- Poncelet's Theorem
- Poncelet Transform
- Poncelet Transverse
- Pong Hau K'i
- Pons Asinorum
- Pontryagin Class
- Pontryagin Duality
- Pontryagin Maximum Principle
- Pontryagin Number
- Ponzo's Illusion
- Pop
- Population Comparison
- Population Growth
- Porism
- Porter's Constant
- Poset
- Poset Dimension
- Position Four-Vector
- Position Vector
- Positive
- Positive Definite Function
- Positive Definite Matrix
- Positive Definite Quadratic Form
- Positive Definite Tensor