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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,
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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