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Recurring Digital Invariant

المؤلف:  Madachy, J. S.

المصدر:  Madachy,s Mathematical Recreations. New York: Dover

الجزء والصفحة:  pp. 163-165

16-11-2020

1041

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

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REFERENCES:

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