Narcissistic number

Definition

Let n be a natural number. We define the narcissistic function for base b 1 F_{b} : \mathbb{N} \rightarrow \mathbb{N} to be the following: : F_{b}(n) = \sum_{i=0}^{k - 1} d_i^k. where k = \lfloor \log_{b}{n} \rfloor + 1 is the number of digits in the number in base b, and : d_i = \frac{n \bmod{b^{i+1}} - n \bmod b^i}{b^i} is the value of each digit of the number. A natural number n is a narcissistic number if it is a fixed point for F_{b}, which occurs if F_{b}(n) = n. The natural numbers 0 \leq n are trivial narcissistic numbers for all b, all other narcissistic numbers are nontrivial narcissistic numbers.

For example, the number 153 in base b = 10 is a narcissistic number, because k = 3 and 153 = 1^3 + 5^3 + 3^3.

A natural number n is a sociable narcissistic number if it is a periodic point for F_{b}, where F_{b}^p(n) = n for a positive integer p (here F_{b}^p is the pth iterate of F_b), and forms a cycle of period p. A narcissistic number is a sociable narcissistic number with p = 1, and an amicable narcissistic number is a sociable narcissistic number with p = 2.

All natural numbers n are preperiodic points for F_{b}, regardless of the base. This is because for any given digit count k, the minimum possible value of n is b^{k - 1}, the maximum possible value of n is b^{k} - 1 \leq b^k, and the narcissistic function value is F_{b}(n) = k(b-1)^k. Thus, any narcissistic number must satisfy the inequality b^{k - 1} \leq k(b-1)^k \leq b^k. Multiplying all sides by \frac{b}{(b - 1)^k}, we get {\left(\frac{b}{b - 1}\right)}^{k} \leq bk \leq b{\left(\frac{b}{b - 1}\right)}^{k}, or equivalently, k \leq {\left(\frac{b}{b - 1}\right)}^{k} \leq bk. Since \frac{b}{b - 1} \geq 1, this means that there will be a maximum value k where {\left(\frac{b}{b - 1}\right)}^{k} \leq bk, because of the exponential nature of {\left(\frac{b}{b - 1}\right)}^{k} and the linearity of bk. Beyond this value k, F_{b}(n) \leq n 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 b^{k} - 1, making it a preperiodic point. Setting b equal to 10 shows that the largest narcissistic number in base 10 must be less than 10^{60}.

The number of iterations i needed for F_{b}^{i}(n) to reach a fixed point is the narcissistic function's persistence of n, and undefined if it never reaches a fixed point.

A base b has at least one two-digit narcissistic number if and only if b^2 + 1 is not prime, and the number of two-digit narcissistic numbers in base b equals \tau(b^2+1)-2, where \tau(n) is the number of positive divisors of n.

Every base b \geq 3 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, ...

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 b. '#' 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

• Joseph S. Madachy, Mathematics on Vacation, Thomas Nelson & Sons Ltd. 1966, pages 163-175.
• Rose, Colin (2005), Radical narcissistic numbers, Journal of Recreational Mathematics, 33(4), 2004–2005, pages 250-254.
• Perfect Digital Invariants by Walter Schneider

References

1. "Narcissistic Number".
2. [http://www.cs.umd.edu/Honors/reports/NarcissisticNums/NarcissisticNums.html ''Perfect and PluPerfect Digital Invariants''] {{webarchive. link. (2007-10-10 by Scott Moore)
3. [https://web.archive.org/web/20091027123639/http://www.geocities.com/~harveyh/narciss.htm PPDI (Armstrong) Numbers] by Harvey Heinz
4. "Armstrong Numbers".
5. Deimel, Lionel. "Mystery Solved!".
6. {{OEIS