“Narcissistic Number”的概念、定义、习题解题方法及翻译-数学辞典

单词

Narcissistic Number

释义

Narcissistic Number

An -Digit number which is the Sum of the th Powers of its Digits iscalled an -narcissistic number, or sometimes an Armstrong Number or Perfect Digital Invariant (Madachy 1979).The smallest example other than the trivial 1-Digit numbers is

The series of smallest narcissistic numbers of

digits are 0, (none), 153, 1634, 54748, 548834, ... (Sloane's A014576). Hardy (1993) wrote, ``There are just four numbers, after unity, which are the sums of the cubes of their digits:

, and

. These are odd facts, very suitable for puzzlecolumns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician.'' The following tablegives the generalization of these ``unappealing'' numbers to other Powers (Madachy 1979, p. 164).

	-narcissistic numbers
1	0, 1, 2, 3, 4, 5, 6, 7, 8, 9
2	none
3	153, 370, 371, 407
4	1634, 8208, 9474
5	54748, 92727, 93084
6	548834
7	1741725, 4210818, 9800817, 9926315
8	24678050, 24678051, 88593477
9	146511208, 472335975, 534494836, 912985153
10	4679307774

A total of 88 narcissistic numbers existin base-10, as proved by D. Winter in 1985 and verified by D. Hoey. These numbers exist for only 1, 3, 4, 5, 6, 7, 8, 9, 10,11, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 29, 31, 32, 33, 34, 35, 37, 38, and 39 digits. It can easily be shown thatbase-10 -narcissistic numbers can exist only for , since

for

.The largest base-10 narcissistic number is the 39-narcissistic

A table of the largest known narcissistic numbers in various Bases is given by Pickover (1995). Atabulation of narcissistic numbers in various bases is given by (Corning).

A closely related set of numbers generalize the narcissistic number to -Digit numbers which are the sums of any single Power of their Digits. For example, 4150 is a 4-Digit number which is the sum offifth Powers of its Digits. Since the number of digits is not equal to the power to which theyare taken for such numbers, it is not a narcissistic number. The smallest numbers which are sums of any singlepositive power of their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151, 8208, 9474, ...(Sloane's A023052), with powers 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 4, 5, 5, 4, 4, ... (Sloane's A046074).

The smallest numbers which are equal to the th powers of their digits for , 4, ..., are 153, 1634, 4150, 548834,1741725, ... (Sloane's A003321). The -digit numbers equal to the sum of th powers of their digits (a finite sequence) arecalled Armstrong Numbers or Plus Perfect Numbers and are given by1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ... (Sloane's A005188).

If the sum-of-th-powers-of-digits operation applied iteratively to a number eventually returns to , the smallestnumber in the sequence is called a -Recurring Digital Invariant.

See also Additive Persistence, Digital Root, Digitaddition, Kaprekar Number, MultiplicativeDigital Root, Multiplicative Persistence, Recurring Digital Invariant, Vampire Number

References

Corning, T. ``Exponential Digital Invariants.'' http://members.aol.com/tec153/Edi4web/Edi.html.

Hardy, G. H. A Mathematician's Apology. New York: Cambridge University Press, p. 105, 1993.

Madachy, J. S. ``Narcissistic Numbers.'' Madachy's Mathematical Recreations. New York: Dover, pp. 163-173, 1979.

Pickover, C. A. Keys to Infinity. New York: Wiley, pp. 169-170, 1995.

Rumney, M. ``Digital Invariants.'' Recr. Math. Mag. No. 12, 6-8, Dec. 1962.

Sloane, N. J. A. Sequences A014576,A023052,A046074,A005188/M0488, andA003321/M5403in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.

Weisstein, E. W. ``Narcissistic Numbers.'' Mathematica notebook Narcissistic.dat.

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