In number theory, a narcissistic number[1][2] (also known as a pluperfect digital invariant (PPDI),[3] an Armstrong number[4] (after Michael F. Armstrong)[5] or a plus perfect number)[6] in a given number base
is a number that is the sum of its own digits each raised to the power of the number of digits.
We define the narcissistic function for base
, and is the value of each digit of the number.
A natural number
is a narcissistic number if it is a fixed point for
are trivial narcissistic numbers for all
, all other narcissistic numbers are nontrivial narcissistic numbers.
A natural number
is a sociable narcissistic number if it is a periodic point for
), and forms a cycle of period
A narcissistic number is a sociable narcissistic number with
, and an amicable narcissistic number is a sociable narcissistic number with
All natural numbers
This is because for any given digit count
Thus, any narcissistic number must satisfy the inequality
, because of the exponential nature of
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
, making it a preperiodic point.
equal to 10 shows that the largest narcissistic number in base 10 must be less than
to reach a fixed point is the narcissistic function's persistence of
, and undefined if it never reaches a fixed point.
has at least one two-digit narcissistic number if and only if
is not prime, and the number of two-digit narcissistic numbers in base
is the number of positive divisors of
that is not a multiple of nine has at least one three-digit narcissistic number.
The bases that do not are There are only 88 narcissistic numbers in base 10, of which the largest is with 39 digits.
[1] All numbers are represented in base
'#' is the length of each known finite sequence.
1234 → 2404 → 4103 → 2323 → 1234 3424 → 4414 → 11034 → 20034 → 20144 → 31311 → 3424 1044302 → 2110314 → 1044302 1043300 → 1131014 → 1043300 44 → 52 → 45 → 105 → 330 → 130 → 44 13345 → 33244 → 15514 → 53404 → 41024 → 13345 14523 → 32253 → 25003 → 23424 → 14523 2245352 → 3431045 → 2245352 12444435 → 22045351 → 30145020 → 13531231 → 12444435 115531430 → 230104215 → 115531430 225435342 → 235501040 → 225435342 Narcissistic numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.