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An -Digit number which is the Sum of the
th Powers of its Digits is
called 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
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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 exist
in 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 that
base-10 -narcissistic numbers can exist only for
, since
A table of the largest known narcissistic numbers in various Bases is given by Pickover (1995). A tabulation 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 of
fifth Powers of its Digits. Since the number of digits is not equal to the power to which they
are taken for such numbers, it is not a narcissistic number. The smallest numbers which are sums of any single
positive 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) are
called Armstrong Numbers or Plus Perfect Numbers and are given by
1, 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 smallest
number in the sequence is called a
-Recurring Digital Invariant.
See also Additive Persistence, Digital Root, Digitaddition, Kaprekar Number, Multiplicative Digital 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, and
A003321/M5403
in ``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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© 1996-9 Eric W. Weisstein