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Recurring Digital Invariant


To define a recurring digital invariant of order k, compute the sum of the kth 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 kth 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^(''),...} is said to be a k-recurring digital invariant. For example,

55:5^3+5^3=250
(1)
250:2^3+5^3+0^3=133
(2)
133:1^3+3^3+3^3=55,
(3)

so 55 is an order 3 recurring digital invariant. The following table gives recurring digital invariants of orders 2 to 10 (Madachy 1979).

orderRDIcycle lengths
248
355, 136, 160, 9193, 2, 3, 2
41138, 21787, 2
5244, 8294, 8299, 9044, 9045, 10933,28, 10, 6, 10, 22, 4, 12, 2, 2
24584, 58618, 89883
617148, 63804, 93531, 239459, 28259530, 2, 4, 10, 3
780441, 86874, 253074, 376762,92, 56, 27, 30, 14, 21
922428, 982108, five more
86822, 7973187, 8616804
9322219, 2274831, 20700388, eleven more
1020818070, five more

See also

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

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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

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