Weird Numbers in C++

In this article, we are going to talk about the Weird Numbers with its history, properties, algorithm, example, advantages, disadvantages, and applications.

What are Weird Numbers?

Weird numbers as numbers of odd integers that are a product of four primes. A number is termed a weird number if it meets two conditions:

• It is an abundant number: The number of proper divisors, which is the total of numbers that divide the number other than the number itself, is greater than the number.
• It is not a semiperfect number: Never can there be found any proper divisor of it that can be together make up the number itself.

For example, the first weird number is 70. It has the proper divisors 1, 2, 5, 7, 10, 14, and 35, and the sum of the divisors equals 74; the number is abundant. However, no subset of these divisors will give a sum of 70. These numbers are rare and computationally interesting in their definition as a boundary between abundant and perfect numbers.

History of Weird Numbers

According to the analysis of the article, weird numbers are numbers that were defined in the set of number theory to look at some other aspect of integers apart from the perfect, deficient and abundant numbers. Numbers with abundant amounts are well researched, but that closure of numbers, which is partly abundant and not semiperfect, has retained mathematical fascination. Researchers soon realized that, as does the y-axis, the stratosphere of the wizened Z’s become even gnarlier, rarer and mathematically more challenging to locate.

Properties of Weird Numbers:

Several properties of Weird Numbers in C++ are as follows:

• Proper divisors: According to the definition, the proper divisors of a number are the numbers by which it must be divided to get an exact quotient without exceeding the number itself.
• Abundance: An integer is an abundant number if the great rational divisor sum of the integer is greater than the integer.
• Non-semi-perfect: If a subset of the proper divisors cannot make the number we are considering, it is not a semi-perfect number.
• Rarity: The appearance of the weird numbers is also rather low, and it decreases as numbers become larger.
• Odd and even: Almost all classes of known weird numbers are even though some constructions allow the possibility of odd weird numbers.

Algorithm to Check for Weird Numbers

To decide whether a number is weird:

Step 1: Identify its proper divisors.

Step 2: If a number is the sum of its proper divisors, the number is said to be an abundant number.

Step 3: In order to check the divisors, we can again use the subset_sum algorithm. If no such subset is found, number is weird.

Example:

Let us take an example to illustrate the Weird Number in C++.

Output:

Enter a number: 65
The given num 65 is not a weird number.   

Explanation:

• The program is written in C++, and its function is to find out if a particular number is a weird number by checking two conditions: the first condition is that the number must be abundant, and the second condition is that it should not equal semiperfect. The first function is called getProperDivisors, which gives proper divisors of the input number num. Proper divisors are those integers that divide num without leaving a remainder, excluding the number itself. This function executes a loop from 1 to num/2 because no divisors will be larger than half of the number except num. Proper divisors are stored in a vector with a return to process later.
• After that, the function isSubSum checks whether the sum of any subset is equal to the number. This check is critical in confirming that a number is truly weird. Bit manipulation is used to find any subsets of divisors. The number of subsets equals 2n, where n is attributed to the size of the divisor vector. The loop traverses through every index bitwise equivalent to the subset index-mask and checks whether the ith bit in the mask is set; that means the divisor corresponding to this is included in the subset-sum. If there's any subset that equals the input number, true is returned, implying that the number is semiperfect; otherwise false.
• The main function combines all the previous concepts to find the weirdity of the number under concern. It calls the getProperDivisors function for the proper divisor, after which the function is used to compute the total of all divisors. If the total of the divisors is less than or equal to the input number, it cannot be weird. Therefore, it can't be abundant.
• If the number succeeds the abundance test, the function calls isSubSum() to determine if any subset of the divisors equals the number itself. In the event of a subset-sum that is valid, it is semiperfect and thus disqualified as weird. The function would return true. Therefore, the number in question is really weird, provided both conditions are met.

Advantages of Weird Numbers:

Several advantages of Weird Numbers in C++ are as follows:

• Theoretical Significance: The study of weird numbers increases the theoretical knowledge of the classification of integers and the abundance and semi-perfect consideration.
• Algorithmic Challenges: Just exploring this class of numbers brings up several computational obstacles, such as the subset-sum problem, which is NP-complete.
• Practical Usage: The study of weird numbers provides avenues for students in studies on divisors, set theory, and manipulation of bits.
• Cryptography: The associations between weird numbers and the subset-sum problem make this particularly significant in terms of cryptography.

Disadvantages of Weird Numbers:

Several disadvantages of Weird Numbers in C++ are as follows:

• Rarity: Weird numbers are very rare, which reduces their functionality in things that matter most.
• Computationally Exhaustive: It is too computationally expensive for larger numbers to determine whether a number is weird, which requires the computation of its divisors and the subset-sum problem.
• Limited Applicability: Although it is of great theoretical magnitude, the application of weird numbers in engineering and science is limited.

Applications of Weird Numbers:

Several applications of Weird Numbers in C++ are as follows:

• The study of weird numbers introduces number theory among properties and classifications of integers.
• They serve as teaching tools for a sequence of mathematical topics that range from divisors to subsets and recursion.
• The design of algorithms: The subset-sum problem of weird numbers is related to problems employed in computer science and cryptography.

Conclusion:

In conclusion, Weird numbers are a niche in the world of Number Theory, combining strange properties of being abundant and non-semi-perfect. A number is classified as a weird number if the sum of its proper divisors exceeds this number, while no subset of those divisors can add up to equal that number (non-semi-perfect). Such numbers are unusual; their study contributes greatly to the field of mathematics and to solving algorithmic challenges, particularly when considering efforts to investigate divisors, subsets, and particularly NP-complete problems like finding a solution for a subset-sum. For example, 70, being the first weird number, represents these two properties beautifully.

The study of weird numbers is interesting and provides deeper insights on how to classify integers based on divisors. Cryptographic and computational problems are also also present. However, because there are few weird numbers and their calculation can be quite complicated even for mathematicians, they still matter more for curiosity and interest. Though they represent these disadvantages, weird numbers are still great uses in teaching mathematical logic, recursion, and manipulating bits—bridging pure maths with computational theory. As more and more research gets done, they are used by mathematicians and computer scientists alike, reflecting the beauty and complexity of mathematical structures.