Wild Narcissistic NumbersMike Keith

In the most general sense, a

narcissistic numberis an integer that's equal to an expression involving its own digits. Often, this term is applied to the specific case of a number equal to a sum of powers of its digits, like the number 153:153 = 1³ + 5³ + 3³

whic is equal to the sum of the cubes of its digits.

What I'm interested in here are

wild narcissistic numbers, in which the expression forn(that also happens to be equal ton) still involves just the digits ofnbut in a more bizarre way. Here's a classic example:362913 = 3!· 6 - F(2) + 9!· 1 - F(3)

where F(

n) is thenth Fibonacci number [F(0) = 0, F(1) = 1, and F(n) = F(n-1) + F(n-2)]. The right hand side consists only of the digits in the number itself,in order(3, 6, 2, 9, 1, 3), but the expression involves addition, subtraction, multiplication, factorials, and Fibonacci numbers!Just thinking about the number of different

kindsof wild narcissistic numbers makes ones head hurt a little. Different mathematical operations could be used (add, subtract, multiply, divide, raise to a power, etc.) as well as various "unary functions" (factorial, subfactorial, etc.), binary functions (for example C(n,m) and P(n,m), the numbers of combinations and permutations ofnthings takenmat a time), and number sequences (like thenth Fibonacci or Catalan number). There are millions of different kinds, even for something as small as 6-digit numbers.Well, I have only explored a tiny grain of sand in this large universe, but here are a few I've come up with. The notation c(

n) represents thenth Catalan number (in the sequence c(0)=1,1,2,5,14,42,132,429,1430,4862,...) and "¡" is the subfactorial function.

36 = 3!· 6

1448 = 1!· 4 + c(4) + c(8)

17812 = 1^{7}+ 8¡ + 1^{2}

24739 = 2^{4}+ 7! + 3^{9}

267499 = 2^{6}+ c(7) + 4^{9}+ c(9)Here are some nice ones sent to me (Apr 2000) by Jean-Chearles Meyrignac:

71 = sqrt(7! + 1)

119 = -(1-({-(1 - sqrt(9)!)}!)

720 = (sqrt(7 + 2 + 0)!)! and also, as pointed out to me by Amnon Melzer, = (7 - 2^{0})!

733 = (7 + 3!) + (3!)!

936 = (sqrt(9)!)^{3}+ 6!