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:

1³ + 5³ + 3³ = 153

To put it algebraically, in a base b, an integer $n = \sum_{i = 1}^k d_ib^{i - 1}$ where $d 1$ is the least significant digit and $d k$ is the most significant, if it's also true that for some m it happens that $n = \sum_{i = 1}^k {d_i}^m$ then n is one of these numbers.

In "A Mathematician's Apology", 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."

However, it is not known if the only base 10 numbers equal to the sum of the cubes of their digits are 1, 153, 370, 371, and 407.

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.

Also see Narcissus (mythology), narcissism.