An n-digit number that is the sum of the nth powers of its digits is called an n-narcissistic number. It is also sometimes known as an Armstrong number, perfect digital invariant (Madachy 1979), or plus perfect number.

The values obtained by summing the dth powers of the digits of a d-digit number n for n = 1, 2, ... are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 5, 10, 17, 26, ... (Sloane's A101337).

The smallest example other than the trivial 1-digit numbers is

The first few are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ... (Sloane's A005188).

The series of smallest narcissistic numbers of n digits are 0, (none), 153, 1634, 54748, 548834, ... (Sloane's A014576). Hardy (1993) wrote, "There are just four numbers, after unity, which are the sums of the cubes of their digits: , , , and . 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." The following table gives the generalization of these "unappealing" numbers to other powers (Madachy 1979, p. 164).

A total of 88 narcissistic numbers exist in base 10, as proved by D. Winter in 1985 and verified by D. Hoey. T. A. Mendes Oliveira é Silva gave the full sequence in a posting (Article 42889) to sci.math on May 9, 1994. These numbers exist for only 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 16, 17, 19, 20, 21, 23, 24, 25, 27, 29, 31, 32, 33, 34, 35, 37, 38, and 39 digits. It can easily be shown that base-10 n-narcissistic numbers can exist only for , since

for n > 60. The largest base-10 narcissistic number is the 39-narcissistic

A table of the largest known narcissistic numbers in various bases is given by Pickover (1995). A tabulation of narcissistic numbers in various bases is given by (Corning).

A closely related set of numbers generalize the narcissistic number to n-digit numbers which are the sums of any single power of their digits. For example, 4150 is a 4-digit number which is the sum of fifth powers of its digits. Since the number of digits is not equal to the power to which they are taken for such numbers, they are not narcissistic numbers. The smallest numbers which are sums of any single positive power of their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 4150, 4151, 8208, 9474, ... (Sloane's A023052), with powers 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 4, 5, 5, 4, 4, ... (Sloane's A046074).

The smallest numbers which are equal to the nth powers of their digits for n = 3, 4, ..., are 153, 1634, 4150, 548834, 1741725, ... (Sloane's A003321). The n-digit numbers equal to the sum of nth powers of their digits (a finite sequence) are called Armstrong numbers or plus perfect number and are given by 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, ... (Sloane's A005188).

If the sum-of-kth-powers-of-digits operation applied iteratively to a number n eventually returns to n, the smallest number in the sequence is called a k-recurring digital invariant.

The numbers that are equal to the sum of consecutive powers of their digits are given by 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798 (Sloane's A032799), e.g.,

Additive Persistence, Digital Root, Digitaddition, Harshad Number, Kaprekar Number, Multiplicative Digital Root, Multiplicative Persistence, Powerful Number, Recurring Digital Invariant, Vampire Number

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