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 89 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.

Programming example

Python

The example below implements the narcissistic function described in the definition above to search for narcissistic functions and cycles in Python.

def ppdif(x, b):
y = x
digit_count = 0
while y > 0:
digit_count = digit_count + 1
y = y // b
total = 0
while x > 0:
total = total + pow(x % b, digit_count)
x = x // b
return total

def ppdif_cycle(x, b):
seen = []
while x not in seen:
seen.append(x)
x = ppdif(x, b)
cycle = []
while x not in cycle:
cycle.append(x)
x = ppdif(x, b)
return cycle
The following Python program determines whether the integer entered is a Narcissistic / Armstrong number or not.
def no_of_digits(num):
i = 0
while num > 0:
num //= 10
i+=1

return i

def required_sum(num):
i = no_of_digits(num)
s = 0

while num > 0:
digit = num % 10
num //= 10
s += pow(digit, i)

return s

num = int(input("Enter number:"))
s = required_sum(num)

if s

num:
print("Armstrong Number")
else:
print("Not Armstrong Number")

Java

The following Java program determines whether the integer entered is a Narcissistic / Armstrong number or not.

import java.util.Scanner;

public class ArmstrongNumber

C#

The following C# program determines whether the integer entered is a Narcissistic / Armstrong number or not.

using System;

public class Program

C

The following C program determines whether the integer entered is a Narcissistic / Armstrong number or not.

#include
#include
#include

int getNumberOfDigits(int n);
bool isArmstrongNumber(int candidate);

int main()

bool isArmstrongNumber(int candidate)

int getNumberOfDigits(int n)

C++

The following C++ program determines whether the Integer entered is a Narcissistic / Armstrong number or not.

#include
#include

bool isNarcissistic(long n)
int main()

Ruby

The following Ruby program determines whether the integer entered is a Narcissistic / Armstrong number or not.
def narcissistic?(value) #1^3 + 5^3 + 3^3 = 1 + 125 + 27 = 153
nvalue = []
nnum = value.to_s
nnum.each_char do , num,
nvalue << num.to_i
end
sum = 0
i = 0
while sum <= value
nsum = 0
nvalue.each_with_index do , num,idx,
nsum += num ** i
end
if nsum

value
return true
else
i += 1
sum += nsum
end
end
return false
end

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


*

{{Classes of natural numbers

Arithmetic dynamics
Base-dependent integer sequences