Narcissistic number

In number theory, a narcissistic number''Perfect and PluPerfect Digital Invariants''
by Scott Moore (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 $b$ is a number that is the sum of its own digits each raised to the power of the number of digits.

Definition

Let $n$ be a natural number. We define the narcissistic function for base $b > 1$ $F_ : \mathbb \rightarrow \mathbb$ to be the following:
: $F_(n) = \sum_^ d_i^k.$
where $k = \lfloor \log_ \rfloor + 1$ is the number of digits in the number in base $b$, and
: $d_i = \frac$
is the value of each digit of the number. A natural number $n$ is a narcissistic number if it is a fixed point for $F_$, which occurs if $F_(n) = n$. The natural numbers $0 \leq n < b$ 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 $F_$, where $F_^p(n) = n$ for a positive integer $p$ (here $F_^p$ 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 $F_$, regardless of the base. This is because for any given digit count $k$, the minimum possible value of $n$ is $b^$, the maximum possible value of $n$ is $b^ - 1 \leq b^k$, and the narcissistic function value is $F_(n) = k(b-1)^k$. Thus, any narcissistic number must satisfy the inequality $b^ \leq k(b-1)^k \leq b^k$. Multiplying all sides by $\frac$, we get $^ \leq bk \leq b^$, or equivalently, $k \leq ^ \leq bk$. Since $\frac \geq 1$, this means that there will be a maximum value $k$ where $^ \leq bk$, because of the exponential nature of $^$ and the linearity of $bk$. Beyond this value $k$, $F_(n) \leq n$ 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 $b^ - 1$, making it a preperiodic point. Setting $b$ equal to 10 shows that the largest narcissistic number in base 10 must be less than $10^$.

The number of iterations $i$ needed for $F_^(n)$ 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.

See also

* 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

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

External links

Armstrong Numbers

Armstrong Numbers in base 2 to 16


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{{Classes of natural numbers

Arithmetic dynamics
Base-dependent integer sequences