Integer expressible as the sum of its own digits each raised to the power of the number of digits
In number theory, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) in a given number base formula_1 is a number that is the sum of its own digits each raised to the power of the number of digits.
Definition.
Let formula_2 be a natural number. We define the narcissistic function for base formula_3 formula_4 to be the following:
 formula_5
where formula_6 is the number of digits in the number in base formula_1, and 
 formula_8
is the value of each digit of the number. A natural number formula_2 is a narcissistic number if it is a fixed point for formula_10, which occurs if formula_11. The natural numbers formula_12 are trivial narcissistic numbers for all formula_1, all other narcissistic numbers are nontrivial narcissistic numbers.
For example, the number 153 in base formula_14 is a narcissistic number, because formula_15 and formula_16.
A natural number formula_2 is a sociable narcissistic number if it is a periodic point for formula_10, where formula_19 for a positive integer formula_20 (here formula_21 is the formula_20th iterate of formula_23), and forms a cycle of period formula_20. A narcissistic number is a sociable narcissistic number with formula_25, and an amicable narcissistic number is a sociable narcissistic number with formula_26.
All natural numbers formula_2 are preperiodic points for formula_10, regardless of the base. This is because for any given digit count formula_29, the minimum possible value of formula_2 is formula_31, the maximum possible value of formula_2 is formula_33, and the narcissistic function value is formula_34. Thus, any narcissistic number must satisfy the inequality formula_35. Multiplying all sides by formula_36, we get formula_37, or equivalently, formula_38. Since formula_39, this means that there will be a maximum value formula_29 where formula_41, because of the exponential nature of formula_42 and the linearity of formula_43. Beyond this value formula_29, formula_45 always. Thus, there are a finite number of narcissistic numbers, and any natural number is guaranteed to reach a periodic point or a fixed point less than formula_46, making it a preperiodic point. Setting formula_1 equal to 10 shows that the largest narcissistic number in base 10 must be less than formula_48.
The number of iterations formula_49 needed for formula_50 to reach a fixed point is the narcissistic function's persistence of formula_2, and undefined if it never reaches a fixed point.
A base formula_1 has at least one two-digit narcissistic number if and only if formula_53 is not prime, and the number of two-digit narcissistic numbers in base formula_1 equals formula_55, where formula_56 is the number of positive divisors of formula_2.
Every base formula_58 that is not a multiple of nine has at least one three-digit narcissistic number. The bases that do not are
2, 72, 90, 108, 153, 270, 423, 450, 531, 558, 630, 648, 738, 1044, 1098, 1125, 1224, 1242, 1287, 1440, 1503, 1566, 1611, 1620, 1800, 1935, ... (sequence in the OEIS)
There are only 88 narcissistic numbers in base 10, of which the largest is
115,132,219,018,763,992,565,095,597,973,971,522,401
with 39 digits.
Narcissistic numbers and cycles of "F""b" for specific "b".
All numbers are represented in base formula_1. '#' is the length of each known finite sequence.
Extension to negative integers.
Narcissistic numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.
References.
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