An -digit number that is the sum of the th powers of its digits is called an -narcissistic number. It is also 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 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 88 narcissistic numbers exist in base 10, as proved 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 base- narcissistic numbers 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 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, they are not narcissistic numbers. 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, ... (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 are the Münchhausen numbers, which are numbers equal to the sum of their digits raised to each digit's power.

The smallest numbers which are equal to the th powers of their digits for , 4, ..., are 153, 1634, 4150, 548834, 1741725, ... (OEIS A003321). The -digit numbers equal to the sum of th powers of their digits (a finite sequence) are called Armstrong numbers or plus perfect number and are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ... (OEIS 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.

The numbers that are equal to the sum of consecutive powers of their digits are given by 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).

REFERENCES:

Corning, T. "Exponential Digital Invariants." https://members.aol.com/tec153/Edi4web/Edi.html

Deimel, L. E. Jr. and Jones, M. T. "Finding Pluperfect Digital Invariants: Techniques, Results and Observations." J. Recr. Math. 14, 97-108, 1981.

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

Heinz, H. "Narcissistic Numbers." https://www.magic-squares.net/narciss.htm.

Keith, M. "Wild Narcissistic Numbers." https://users.aol.com/s6sj7gt/mikewild.htm.

Lamoitier, J. P. "Fifty Basic Exercises." SYBEX Inc., 1981.

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.

Pickover, C. A. "The Latest Gossip on Narcissistic Numbers." Ch. 88 in Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. Oxford, England: Oxford University Press, pp. 204-205, 2001.

Rivera, C. "Problems & Puzzles: Puzzle 015-Narcissistic and Handsome Primes." https://www.primepuzzles.net/puzzles/puzz_015.htm.

Roberts, J. The Lure of the Integers. Washington, DC: Math. Assoc. Amer., p. 35, 1992.

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

Sloane, N. J. A. Sequences A005188/M0488, A003321/M5403, A010344, A010346, A010348, A010350, A010353, A010354, A014576, A023052, A032799, A046074, A101337, and A114904 in "The On-Line Encyclopedia of Integer Sequences."

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