Narcissistic number explained

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 _{b} : N → N to be the following:F _{b} (n) = \sum ^{k -1} _{i =0} d ^{k.} _{i} where k = \lfloor log _{b} {n} \rfloor + 1 is the number of digits in the number in base b , and d _{i} = (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 < 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 _{b} , where F ^{p(n)} _{b} = n for a positive integer p (here F ^{p} _{b} 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} , 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} \leq k(b -1 ) ^{k} \leq b ^{k} . Multiplying all sides by (b)/((b -1 ) ^{k}) , we get {\left( (b)/(b -1) \right)} ^{k} \leq bk \leq b{\left( (b)/(b -1) \right)} ^{k} , or equivalently, k \leq {\left( (b)/(b -1) \right)} ^{k} \leq bk . Since (b)/(b -1) \geq 1 , this means that there will be a maximum value k where {\left( (b)/(b -1) \right)} ^{k} \leq bk , because of the exponential nature of {\left( (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 ^{i} _{b} (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.

b	Narcissistic numbers		Cycles	OEIS sequence(s)
	0, 1	2	\varnothing
	0, 1, 2, 12, 22, 122	6	\varnothing
	0, 1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303	12	\varnothing	and
	0, 1, 2, 3, 4, 23, 33, 103, 433, 2124, 2403, 3134, 124030, 124031, 242423, 434434444, ...	18	1234 → 2404 → 4103 → 2323 → 1234 3424 → 4414 → 11034 → 20034 → 20144 → 31311 → 3424 1044302 → 2110314 → 1044302 1043300 → 1131014 → 1043300
	0, 1, 2, 3, 4, 5, 243, 514, 14340, 14341, 14432, 23520, 23521, 44405, 435152, 5435254, 12222215, 555435035 ...	31	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
	0, 1, 2, 3, 4, 5, 6, 13, 34, 44, 63, 250, 251, 305, 505, 12205, 12252, 13350, 13351, 15124, 36034, 205145, 1424553, 1433554, 3126542, 4355653, 6515652, 125543055, ...	60
	0, 1, 2, 3, 4, 5, 6, 7, 24, 64, 134, 205, 463, 660, 661, 40663, 42710, 42711, 60007, 62047, 636703, 3352072, 3352272, ...	63		and
	0, 1, 2, 3, 4, 5, 6, 7, 8, 45, 55, 150, 151, 570, 571, 2446, 12036, 12336, 14462, 2225764, 6275850, 6275851, 12742452, ...	59
	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, ...	89
	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, 56, 66, 105, 307, 708, 966, A06, A64, 8009, 11720, 11721, 12470, ...	135
	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 25, A5, 577, 668, A83, 14765, 938A4, 369862, A2394A, ...	88
	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, 14, 36, 67, 77, A6, C4, 490, 491, 509, B85, 3964, 22593, 5B350, ...	202
	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, 136, 409, 74AB5, 153A632, ...	103
	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, 78, 88, C3A, D87, 1774, E819, E829, 7995C, 829BB, A36BC, ...	203
	0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 156, 173, 208, 248, 285, 4A5, 5B0, 5B1, 60B, 64B, 8C0, 8C1, 99A, AA9, AC3, CA8, E69, EA0, EA1, B8D2, 13579, 2B702, 2B722, 5A07C, 5A47C, C00E0, C00E1, C04E0, C04E1, C60E7, C64E7, C80E0, C80E1, C84E0, C84E1, ...	294

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 cycleThe 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.

1. include
2. include
3. 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.

1. include
2. include

bool isArmstrong(int 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 endreturn falseend

JavaScript

The following JavaScript program determines whether the integer entered is a Narcissistic / Armstrong number or not.function narcissistic(number)

Rust

The following Rust program prints all the Narcissistic / Armstrong numbers from 0 to 100 million in base 10.fn is_armstrong_number(num: u64) -> bool

fn main

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

• Digital Invariants
• Armstrong Numbers
• Armstrong Numbers in base 2 to 16
• Armstrong numbers between 1-999 calculator
• Web site: Symonds. Ria. 153 and Narcissistic Numbers. https://ghostarchive.org/varchive/youtube/20211219/4aMtJ-V26Z4 . 2021-12-19 . live. Numberphile. Brady Haran.

Notes and References

1. http://www.cs.umd.edu/Honors/reports/NarcissisticNums/NarcissisticNums.html Perfect and PluPerfect Digital Invariants
2. https://web.archive.org/web/20091027123639/http://www.geocities.com/~harveyh/narciss.htm PPDI (Armstrong) Numbers
3. http://www.deimel.org/rec_math/dik1.htm Armstrong Numbers
4. http://blog.deimel.org/2010/05/mystery-solved.html Lionel Deimel’s Web Log