Recurring Digital Invariant

To define a recurring digital invariant of order $k$, compute the sum of the $k$th powers of the digits of a number $n$. If this number $n'$ is equal to the original number $n$, then $n=n'$ is called a $k$-Narcissistic Number. If not, compute the sums of the $k$th powers of the digits of $n'$, and so on. If this process eventually leads back to the original number $n$, the smallest number in the sequence $\{n, n', n'', \ldots\}$ is said to be a $k$-recurring digital invariant. For example,

$$55: 5^3+5^3 = 250$$

$$250: 2^3+5^3+0^3 = 133$$

$$133: 1^3+3^3+3^3 = 55,$$

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

See also 196-Algorithm, Additive Persistence, Digitaddition, Digital Root, Happy Number, Kaprekar Number, Narcissistic Number, Vampire Number

References

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 163-165, 1979.