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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,

$\displaystyle 55: 5^3+5^3$ $\textstyle =$ $\displaystyle 250$  
$\displaystyle 250: 2^3+5^3+0^3$ $\textstyle =$ $\displaystyle 133$  
$\displaystyle 133: 1^3+3^3+3^3$ $\textstyle =$ $\displaystyle 55,$  

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.



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© 1996-9 Eric W. Weisstein
1999-05-25