Narcissistic number

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In number theory, a narcissistic number or pluperfect digital invariant (PPDI) or Armstrong number is a number that in a given base is the sum of its own digits to the power of the number of digits.

For example, the decimal (Base 10) number 153 is an Armstrong number, because:

1³ + 5³ + 3³ = 153

To put it algebraically, let $n = \sum_{i = 1}^k d_ib^{i - 1}$ be an integer with representation $d k d k - 1 ... d 1$ in base-b notation. If for some m it happens that $n = \sum_{i = 1}^k {d_i}^m$ then n is a narcissistic number (or an m-narcissistic number).

In "A Mathematician's Apology", G. H. Hardy 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."

1 Armstrong numbers in alternate bases
2 See also
3 References
4 External links

[edit] Armstrong numbers in alternate bases

Some "base ten" Armstrong numbers are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, ... (sequence A005188 in OEIS)

Some "base three" Armstrong numbers are: 0,1,2,12,122

Some "base four" Armstrong numbers are: 0,1,2,3,313

[edit] See also

Sometimes the term "narcissistic number" is used to mean any kind of number that is representable by mathematical manipulation of the digits of the number itself. These include: constant base numbers, noteworthy numbers, ascending and descending powers, wild narcissistic numbers, amicable pairs, power-sum numbers, Brown numbers, perfect digital invariant numbers, Friedman numbers, sum-product numbers, recurring digital invariant numbers, happy numbers, etc.

The concept of a narcissistic number can be extended naturally to four-digit numbers (digits raised to the fourth power), five-digit numbers (fifth power), and so on. However, there are a finite number of such numbers, because at a certain point, n x 9^n < 10^(n-1) - 1.

Also see Narcissus (mythology), narcissism.