Last Updated on June 1, 2023 by Mayank Dham
In number theory, an abundant number is a positive integer that is smaller than the sum of its proper divisors. In other words, the sum of the divisors of an abundant number exceeds the number itself. In this article, we will explore the concept of abundant numbers and provide Python code examples to check if a given number is abundant.
Understanding the Abundant Number in Depth
An abundant number, n, can be expressed as the sum of its proper divisors (excluding n itself). If the sum of the divisors is greater than n, the number is classified as abundant. For example, let’s consider the number 12. Its proper divisors are 1, 2, 3, 4, and 6, with a sum of 16. Since 16 is greater than 12, 12 is an abundant number.
We can solve this problem using two methods :
 Using range until number
 Using range until sqrt()
Algorithm to Check Abundant Number using range until Number
 Initialize a variable,
num
, with the number to be checked for abundance.  Initialize a variable,
divisor_sum
, to 0.  Iterate from 1 to
num // 2
(inclusive) using a for loop.  Check if the current number is a divisor of
num
by using the modulo operator (num % i == 0
).  If the current number is a divisor, add it to
divisor_sum
.  After the loop, compare
divisor_sum
withnum
. Ifdivisor_sum
is greater thannum
, the number is abundant; otherwise, it is not.  Return the result (either true or false), indicating whether the number is abundant or not.
Python code to check if a number is abundant using range until Number
n = 12 sum=1 for i in range (2, n): if(n%i==0): sum=sum+i if(sum>n): print(n,'is Abundant Number') else: print(n,'is not Abundant Number')
Output
12 is Abundant Number
Algorithm to Check Abundant Number using range until sqrt()
 Import the
math
module to access mathematical functions.  Set the value of
n
as the number to be checked for abundance.  Initialize a variable
sum
as 1, since 1 can divide any number.  Initialize a variable
i
as 2.  Enter a while loop with the condition
i <= math.sqrt(n)
.  Inside the while loop:
 Check if
n
is divisible byi
using the modulo operator (n % i == 0
).  If
n
is divisible byi
, proceed to the next step.  Check if the quotient (
n // i
) is equal to the divisor (i
).  If the quotient is equal to the divisor, add only one of them to the
sum
.  If the quotient is not equal to the divisor, add both the divisor and quotient to the
sum
.  Increment
i
by 1.
 Check if
 After the while loop:
 Check if the
sum
is greater thann
.  If
sum
is greater thann
, print thatn
is an abundant number.  If
sum
is not greater thann
, print thatn
is not an abundant number.
 Check if the
Python code to check if a number is abundant using range until sqrt()
import math n = 12 sum=1 # 1 can divide any number i=2 while(i<=math.sqrt(n)): if(n%i==0): #if number is divisible by i add the number if(n//i==i): # if quotient is equal to divisor add only one of them sum=sum+i else: sum=sum + i + n/i i=i+1 if(sum>n): print(n,"is Abundant Number") else: print(n,"is not Abundant Number")
Output
12 is Abundant Number
Conclusion
Determining whether a number is abundant can be done by calculating the sum of its proper divisors and comparing it with the number itself. In this article, we discussed the concept of abundant numbers and provided a Python algorithm to check if a number is abundant. By utilizing the code example and the is_abundant
function, you can easily identify and work with abundant numbers in your Python programs.
Frequently Asked Questions (FAQs)

What is the significance of identifying abundant numbers in programming?
Identifying abundant numbers can be useful in various mathematical and computational tasks, such as number theory problems, cryptography, algorithm design, and optimization techniques. 
Can you provide some examples of abundant numbers?
Sure! Examples of abundant numbers include 12, 18, 20, 24, and 30. These numbers have the property that the sum of their proper divisors is greater than the number itself. 
Is it necessary to check all divisors up to
num // 2
in the algorithm?
No, it is not necessary to check all divisors up tonum // 2
. Checking up to the square root ofnum
(i.e.,math.sqrt(num)
) is sufficient to find all divisors. This optimization reduces the number of iterations in the loop. 
Are there any known relationships or patterns among abundant numbers?
Abundant numbers are a subject of study in number theory. They exhibit interesting relationships with other number properties, such as perfect numbers and deficient numbers. However, no general pattern has been found to generate all the abundant numbers. 
How can I utilize the concept of abundant numbers in realworld applications?
Abundant numbers have applications in diverse fields, including cryptography, data analysis, optimization problems, and algorithm design. For example, in cryptography, abundant numbers can be utilized for developing secure encryption algorithms.
.