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narcissistic numbers
A number  $n$  of  $k$  digits is called narcissistic if it is equal to the sum of the  $k$-th powers of its digits.

For example,  $153$  is narcissistic because  $153 = 1^3+5^3+3^3$.

Narcissistic numbers are also called Armstrong or plus-perfect numbers.

D. Winter proved and D.Hoey verified that there are exacly 88 narcissistic numbers. They are 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153, 4679307774, 32164049650, 32164049651, 40028394225, 42678290603, 44708635679, 49388550606, 82693916578, 94204591914, 28116440335967, 4338281769391370, 4338281769391371, 21897142587612075, 35641594208964132, 35875699062250035, 1517841543307505039, 3289582984443187032, 4498128791164624869, 4929273885928088826, 63105425988599693916, 128468643043731391252, 449177399146038697307, 21887696841122916288858, 27879694893054074471405, 27907865009977052567814, 28361281321319229463398, 35452590104031691935943, 174088005938065293023722, 188451485447897896036875, 239313664430041569350093, 1550475334214501539088894, 1553242162893771850669378, 3706907995955475988644380, 3706907995955475988644381, 4422095118095899619457938, 121204998563613372405438066, 121270696006801314328439376, 128851796696487777842012787, 174650464499531377631639254, 177265453171792792366489765, 14607640612971980372614873089, 19008174136254279995012734740, 19008174136254279995012734741, 23866716435523975980390369295, 1145037275765491025924292050346, 1927890457142960697580636236639, 2309092682616190307509695338915, 17333509997782249308725103962772, 186709961001538790100634132976990, 186709961001538790100634132976991, 1122763285329372541592822900204593, 12639369517103790328947807201478392, 12679937780272278566303885594196922, 1219167219625434121569735803609966019, 12815792078366059955099770545296129367, 115132219018763992565095597973971522400, and 115132219018763992565095597973971522401

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of narcissistic numbers

A graph displaying how many narcissistic numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of narcissistic numbers
Useful links Mathworld, Narcissistic Number
Wikipedia, Narcissistic number
OEIS, Sequence A005188
Narcissistic numbers can also be... (you may click on names or numbers and on + to get more values)

aban 153 370 371 407 abundant 8208 9474 93084 4210818 admirable 9474 alternating 1634 9474 92727 amenable 153 8208 54748 93084 1741725 9800817 88593477 146511208 534494836 912985153 apocalyptic 1634 8208 9474 arithmetic 153 371 407 1634 8208 9474 93084 548834 4210818 9800817 9926315 binomial 153 brilliant 407 canyon 407 8208 congruent 371 407 54748 92727 93084 1741725 Curzon 153 548834 cyclic 371 407 9926315 d-powerful 153 370 371 407 1634 8208 9474 54748 92727 93084 + 1741725 4210818 9800817 9926315 decagonal 370 deficient 153 370 371 407 1634 54748 92727 548834 1741725 9800817 9926315 dig.balanced 153 1634 146511208 Duffinian 371 407 92727 1741725 economical 371 equidigital 371 evil 153 371 407 8208 9474 54748 92727 93084 1741725 4210818 24678050 146511208 Friedman 153 gapful 1741725 472335975 Harshad 153 370 407 8208 9926315 88593477 472335975 hexagonal 153 hoax 1741725 24678050 iban 370 371 407 inconsummate 371 interprime 370 junction 24678050 lucky 92727 magnanimous 370 407 modest 92727 Moran 153 370 407 9926315 mountain 153 370 371 oban 370 odious 370 1634 548834 9800817 9926315 24678051 88593477 472335975 534494836 912985153 pancake 407 pandigital 1634 pernicious 370 1634 8208 548834 9800817 practical 8208 prim.abundant 9474 pseudoperfect 8208 9474 93084 Ruth-Aaron 153 370 self 92727 548834 472335975 semiprime 371 407 24678051 sphenic 370 1634 9474 9926315 tau 93084 triangular 153 Ulam 370 9800817 unprimeable 54748 wasteful 153 370 407 1634 8208 9474 54748 92727 93084 548834 1741725 4210818 9800817 9926315 Zumkeller 8208 9474 93084