Narcissistic Number

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

$$153 = 1^3+5^3+3^3.$$

The series of smallest narcissistic numbers of $n$ 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: $153=1^3+5^3+3^3$, $370=3^3+7^3+0^3$, $371=3^3+7^3+1^3$, and $407=4^3+0^3+7^3$. These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician.'' The following table gives the generalization of these ``unappealing'' numbers to other Powers (Madachy 1979, p. 164).

$n$	$n$-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 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 $n$-narcissistic numbers can exist only for $n\leq 60$, since

$$n\cdot 9^n<10^{n-1}$$

for $n>60$. The largest base-10 narcissistic number is the 39-narcissistic

$$115132219018763992565095597973971522401$$

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 $n$-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 $n$th powers of their digits for $n=3$, 4, ..., are 153, 1634, 4150, 548834, 1741725, ... (Sloane's A003321). The $n$-digit numbers equal to the sum of $n$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-$k$th-powers-of-digits operation applied iteratively to a number $n$ eventually returns to $n$, the smallest number in the sequence is called a $k$-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.