Narcissistic Number - Definition

Definition

The definition of a narcissistic number relies on the decimal representation n = d_kd_k-1...d₁d₀ of a natural number n, e.g.

n = d_k·10k-1 + d_k-1·10k-2 + ... + d₂·10 + d₁,

with k digits d_i satisfying 0 ≤ d_i ≤ 9. Such a number n is called narcissistic if it satisfies the condition

n = d_kk + d_k-1k + ... + d₂k + d₁k.

For example the 3-digit decimal number 153 is a narcissistic number because 153 = 13 + 53 + 33.

Narcissistic numbers can also be defined with respect to numeral systems with a base b other than b = 10. The base-b representation of a natural number n is defined by

n = d_kbk-1 + d_k-1bk-2 + ... + d₂b + d₁,

where the base-b digits d_i satisfy the condition 0 ≤ d_i ≤ b-1. For example the (decimal) number 17 is a narcissistic number with respect to the numeral system with base b = 3. Its three base-3 digits are 122, because 17 = 1·32 + 2·3 + 2, and it satisfies the equation 17 = 13 + 23 + 23.

If the constraint that the power must equal the number of digits is dropped, so that for some m possibly different from k it happens that

n = d_km + d_k-1m + ... + d₂m + d₁m,

then n is called a perfect digital invariant or PDI. For example, the decimal number 4150 has four decimal digits and is the sum of the fifth powers of its decimal digits

4150 = 45 + 15 + 55 + 05,

so it is a perfect digital invariant but not a 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:

.

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.