TOPICS

# 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

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

## Explore with Wolfram|Alpha

More things to try:

## References

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

## Referenced on Wolfram|Alpha

Recurring Digital Invariant

## Cite this as:

Weisstein, Eric W. "Recurring Digital Invariant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RecurringDigitalInvariant.html