narcissistic number

Let $n$ be a natural number. We define the narcissistic function for base $b\; >\; 1$ $F\_\{b\}\; :\; \backslash mathbb\{N\}\; \backslash rightarrow\; \backslash mathbb\{N\}$ to be the following:

: $F\_\{b\}(n)\; =\; \backslash sum\_\{i=0\}^\{k\; -\; 1\}\; d\_i^k.$

where $k\; =\; \backslash lfloor\; \backslash log\_\{b\}\{n\}\; \backslash rfloor\; +\; 1$ is the number of digits in the number in base $b$, and

: $d\_i\; =\; \backslash frac\{n\; \backslash bmod\{b^\{i+1\}\}\; -\; n\; \backslash bmod\; b^i\}\{b^i\}$

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 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 $p$th 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 $b^\{k\}\; -\; 1\; \backslash leq\; b^k$, and the narcissistic function value is $F\_\{b\}(n)\; =\; k(b-1)^k$. Thus, any narcissistic number must satisfy the inequality $b^\{k\; -\; 1\}\; \backslash leq\; k(b-1)^k\; \backslash leq\; b^k$. Multiplying all sides by $\backslash frac\{b\}\{(b\; -\; 1)^k\}$, we get $\{\backslash left(\backslash frac\{b\}\{b\; -\; 1\}\backslash right)\}^\{k\}\; \backslash leq\; bk\; \backslash leq\; b\{\backslash left(\backslash frac\{b\}\{b\; -\; 1\}\backslash right)\}^\{k\}$, or equivalently, $k\; \backslash leq\; \{\backslash left(\backslash frac\{b\}\{b\; -\; 1\}\backslash right)\}^\{k\}\; \backslash leq\; bk$. Since $\backslash frac\{b\}\{b\; -\; 1\}\; \backslash geq\; 1$, this means that there will be a maximum value $k$ where $\{\backslash left(\backslash frac\{b\}\{b\; -\; 1\}\backslash right)\}^\{k\}\; \backslash leq\; bk$, because of the exponential nature of $\{\backslash left(\backslash frac\{b\}\{b\; -\; 1\}\backslash right)\}^\{k\}$ and the linearity of $bk$. Beyond this value $k$, $F\_\{b\}(n)\; \backslash leq\; n$ 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 $\backslash tau(b^2+1)-2$, where $\backslash tau(n)$ is the number of positive divisors of $n$.

Every base $b\; \backslash geq\; 3$ 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, ... {{OEIS|id=A248970}}

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.

All numbers are represented in base $b$. '#' is the length of each known finite sequence.

Narcissistic numbers can be extended to the negative integers by use of a signed-digit representation to represent each integer.

• Arithmetic dynamics
• Dudeney number
• Factorion
• Happy number
• Kaprekar's constant
• Kaprekar number
• Meertens number
• Perfect digit-to-digit invariant
• Perfect digital invariant
• Sum-product number

{{reflist}}

{{refbegin}}

• 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.
• [https://web.archive.org/web/20040606020332/http://mathews-archive.com/digit-related-numbers/pdi.html Perfect Digital Invariants] by Walter Schneider

{{refend}}

• [http://www.deimel.org/rec_math/DI_0.htm Digital Invariants]
• [https://web.archive.org/web/20171228054132/https://everything2.net/index.pl?node_id=1407017&displaytype=printable&lastnode_id=1407017 Armstrong Numbers]
• [https://web.archive.org/web/20100109234250/http://ftp.cwi.nl/dik/Armstrong Armstrong Numbers in base 2 to 16]
• [http://www.cs.mtu.edu/~shene/COURSES/cs201/NOTES/chap04/arms.html Armstrong numbers between 1-999 calculator]
• {{cite web|last=Symonds|first=Ria|title=153 and Narcissistic Numbers|url=https://www.youtube.com/watch?v=4aMtJ-V26Z4 |archive-url=https://ghostarchive.org/varchive/youtube/20211219/4aMtJ-V26Z4 |archive-date=2021-12-19 |url-status=live|work=Numberphile|date=3 January 2012 |publisher=Brady Haran}}{{cbignore}}

{{Classes of natural numbers}}

Category:Arithmetic dynamics

Category:Base-dependent integer sequences