A narcissistic number is a nonnegative integer with 
decimal digits equal to the sum
of the 
th
powers of its digits. It is also
called an 
-narcissistic
number and is sometimes known as an Armstrong number, perfect digital invariant (Madachy
1979), or plus perfect number. 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 puzzle
columns and likely to amuse amateurs, but there is nothing in them which appeals
to the mathematician." Narcissistic numbers therefore generalize these "unappealing"
numbers to other powers (Madachy 1979, p. 164).
The smallest example of a narcissistic number other than the trivial 1-digit numbers is
 |
(1)
|
The first few are given by 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ... (OEIS A005188).
It can easily be shown that base-10 
-narcissistic numbers can exist only for 
, since
 |
(2)
|
for 
.
In fact, as summarized in the table below, a total of 89 narcissistic numbers, including
0, exist in base 10. The 88 positive values were found by
D. Winter in 1985 and verified by D. Hoey. T. A. Mendes Oliveira
e Silva gave the full sequence in a posting (Article 42889) to sci.math
on May 9, 1994. 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 (OEIS
A114904) digits,
and the series of smallest narcissistic numbers of 
digits are 0, (none), 153, 1634,
54748, 548834, ... (OEIS A014576).
 | base-10  -narcissistic numbers |
| 1 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 |
| 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 |
| 11 | 32164049650, 32164049651, 40028394225, 42678290603, 44708635679, 49388550606, 82693916578, 94204591914 |
| 14 | 28116440335967 |
| 16 | 4338281769391370, 4338281769391371 |
| 17 | 21897142587612075, 35641594208964132, 35875699062250035 |
| 19 | 1517841543307505039, 3289582984443187032, 4498128791164624869, 4929273885928088826 |
| 20 | 63105425988599693916 |
| 21 | 128468643043731391252, 449177399146038697307 |
| 23 | 21887696841122916288858, 27879694893054074471405, 27907865009977052567814, 28361281321319229463398, 35452590104031691935943 |
| 24 | 174088005938065293023722, 188451485447897896036875, 239313664430041569350093 |
| 25 | 1550475334214501539088894, 1553242162893771850669378, 3706907995955475988644380, 3706907995955475988644381, 4422095118095899619457938 |
| 27 | 121204998563613372405438066, 121270696006801314328439376, 128851796696487777842012787, 174650464499531377631639254, 177265453171792792366489765 |
| 29 | 14607640612971980372614873089, 19008174136254279995012734740, 19008174136254279995012734741, 23866716435523975980390369295 |
| 31 | 1145037275765491025924292050346, 1927890457142960697580636236639, 2309092682616190307509695338915 |
| 32 | 17333509997782249308725103962772 |
| 33 | 186709961001538790100634132976990, 186709961001538790100634132976991 |
| 34 | 1122763285329372541592822900204593 |
| 35 | 12639369517103790328947807201478392, 12679937780272278566303885594196922 |
| 37 | 1219167219625434121569735803609966019 |
| 38 | 12815792078366059955099770545296129367 |
| 39 | 115132219018763992565095597973971522400, 115132219018763992565095597973971522401 |
The table below gives the first few positive base-
narcissistic numbers, written in base
10, for small bases 
. A table of the largest known narcissistic numbers in various
bases is given by Pickover (1995) and a tabulation of narcissistic
numbers in various bases is given by Corning.
 | OEIS | base- narcissistic numbers |
| 2 | 1 | |
| 3 | 1, 2, 5, 8, 17 | |
| 4 | A010344 | 1, 2, 3, 28, 29, 35, 43, 55, 62, 83, 243 |
| 5 | A010346 | 1, 2, 3, 4, 13, 18, 28, 118, 289, 353, 419, 4890, 4891, 9113, ... |
| 6 | A010348 | 1, 2, 3, 4, 5, 99, 190, 2292, 2293, 2324, 3432, 3433, 6197, ... |
| 7 | A010350 | 1, 2, 3, 4, 5, 6, 10, 25, 32, 45, 133, 134, 152, 250, 3190, ... |
| 8 | A010354 | 1, 2, 3, 4, 5, 6, 7, 20, 52, 92, 133, 307, 432, 433, ... |
| 9 | A010353 | 1, 2, 3, 4, 5, 6, 7, 8, 41, 50, 126, 127, 468, ... |
A closely related set generalizes narcissistic numbers to positive integers which are the sums of any single positive power of their decimal digits. For example, 4150 is a 4-digit number which is the sum of fifth powers of its digits. This set includes all positive narcissistic numbers, but 4150 is not narcissistic because its digits are raised to the fifth rather than the fourth power. The smallest positive integers 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, ... (OEIS A023052), with powers 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 4, 5, 5, 4, 4, ... (OEIS A046074).
Another set of related numbers consists of the Münchhausen numbers, which are numbers equal to the sum of their digits raised to each digit's power.
The smallest numbers greater than 1 which are equal to the sum of the 
th
powers of their decimal digits
for 
,
4, ... are 153, 1634, 4150, 548834, 1741725, ... (OEIS A003321).
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.
The numbers that are equal to the sum of consecutive powers of their decimal digits, starting with the first power at the leftmost digit, begin with 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798, ... (OEIS A032799), e.g.,
 |
(3)
|
The values obtained by summing the 
th powers of the digits
of a 
-digit number 
for 
, 2, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 10, 17,
26, ... (OEIS A101337).
