Proper Divisors Research Articles

Imperative Genetic Programming

Genetic programming (GP) has a long-standing tradition in the evolution of computer programs, predominantly utilizing tree and linear paradigms, each with distinct advantages and limitations. Despite the rapid growth of the GP field, there have been disproportionately few attempts to evolve ’real’ Turing-like imperative programs (as contrasted with functional programming) from the ground up. Existing research focuses mainly on specific special cases where the structure of the solution is partly known. This paper explores the potential of integrating tree and linear GP paradigms to develop an encoding scheme that universally supports genetic operators without constraints and consistently generates syntactically correct Python programs from scratch. By blending the symmetrical structure of tree-based representations with the inherent asymmetry of linear sequences, we created a versatile environment for program evolution. Our approach was rigorously tested on 35 problems characterized by varying Halstead complexity metrics, to delineate the approach’s boundaries. While expected brute-force program solutions were observed, our method yielded more sophisticated strategies, such as optimizing a program by restricting the division trials to the values up to the square root of the number when counting its proper divisors. Despite the recent groundbreaking advancements in large language models, we assert that the GP field warrants continued research. GP embodies a fundamentally different computational paradigm, crucial for advancing our understanding of natural evolutionary processes.

A Perfect Infinite Series

Summary We show that the sum of the series ∑ n = 1 ∞ ( n 2 / 2 n ) is the perfect number 6, using simpler series with sums 1, 2, 3, which are the proper divisors of 6.

Application of Partitions of Odd Numbers and their Odd Sums to Prove the Nonexistence of Odd Perfect Numbers

Perfect numbers, which are integers equal to the sum of their proper divisors, excluding themselves, have intrigued mathematicians for centuries. While it is established that even perfect numbers can be expressed as 2p-1(2p-1), where p and 2p-1 are prime numbers (Mersenne primes), the existence of odd perfect numbers remains an unsolved problem. This study aims to prove the nonexistence of odd perfect numbers by utilizing an algorithm which demonstrates that a positive even integer can be partitioned into all pairs of odd numbers. Using this approach, it is proven that any positive odd number 2n+1 can be partitioned into all pairs of both odd and even numbers and from the set of these partitions, we show that there exist a proper subset containing all proper divisors of 2n+1. Using these results, and the facts that there exist infinitely many odd numbers and the odd sums of odd numbers is always odd, we prove the nonexistence of odd perfect numbers contributing to the conjecture that they do not exist.

On Planarity and the Metric Dimension of Proper Divisor Graph of Positive Integers

On Planarity and the Metric Dimension of Proper Divisor Graph of Positive Integers

Wiener index of an ideal-based zero-divisor graph of commutative ring with unity

The Wiener index of a connected graph G is W ( G ) = ∑ { u , v } ⊆ V ( G ) d G ( u , v ) . In this paper, we obtain the Wiener index of H-generalized join of graphs G 1 , G 2 , … , G k . As a consequence, we obtain some earlier known results in [Alaeiyan et al. in Aust. J. Basic Appl. Sci. (2011) 5(12): 145–152; Yeh et al. in Discrete Math. (1994) 135: 359–365] and we also obtain the Wiener index of the generalized corona product of graphs. We further show that the ideal-based zero-divisor graph Γ I ( R ) is a H-generalized join of complete graphs and totally disconnected graphs. As a result, we find the Wiener index of the ideal-based zero-divisor graph Γ I ( R ) and we deduce some of the main results in [Selvakumar et al. in Discrete Appl. Math. (2022) 311: 72–84]. Moreover, we show that W ( Γ I ( Z n ) ) is a quadratic polynomial in n, where Z n is the ring of integers modulo n and we calculate the exact value of the Wiener index of Γ Nil ( R ) ( R ) , where Nil(R) is nilradical of R. Furthermore, we give a Python program for computing the Wiener index of Γ I ( Z n ) if I is an ideal of Z n generated by pr , where pr is a proper divisor of n, p is a prime number and r is a positive integer with r ≥ 2 .

Recursively divisible numbers

We introduce and study the recursive divisor function, a recursive analog of the usual divisor function: κx(n)=nx+∑d⌊nκx(d), where the sum is over the proper divisors of n. We give a geometrical interpretation of κx(n), which we use to derive a relation between κx(n) and κ0(n). For x≥2, we observe that κx(n)/nx<1/(2−ζ(x)). We show that, for n≥2, κ0(n) is twice the number of ordered factorizations, a problem much studied in its own right. By computing those numbers that are more recursively divisible than all of their predecessors, we recover many of the numbers prevalent in design and technology, and suggest new ones which have yet to be adopted.

Realization of zero-divisor graphs of finite commutative rings as threshold graphs

Let R be a finite commutative ring with unity, and let $$G = (V, E)$$ be a simple graph. The zero-divisor graph, denoted by $$\Gamma (R)$$ is a simple graph with vertex set as R, and two vertices $$x, y \in R$$ are adjacent in $$\Gamma (R)$$ if and only if $$xy=0$$ . In [5], the authors have studied the Laplacian eigenvalues of the graph $$\Gamma (\mathbb {Z}_{n})$$ and for distinct proper divisors $$d_1, d_2, \cdots , d_k$$ of n, they defined the sets as, $$\mathcal {A}_{d_i} = \{x \in \mathbb {Z}_{n} : (x, n) = d_i\}$$ , where (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets $$\mathcal {A}_{d_i}$$ , $$1 \le i \le k$$ are actually orbits of the group action: $$Aut(\Gamma (R)) \times R \longrightarrow R$$ , where $$Aut(\Gamma (R))$$ denotes the automorphism group of $$\Gamma (R)$$ . Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that $$\Gamma (R)$$ is a connected threshold graph if and only if $$R\cong {F}_{q}$$ or $$R\cong {F}_2 \times {F}_{q}$$ . We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold.

Sums of proper divisors follow the Erdős–Kac law

Let s ( n ) = ∑ d ∣ n , d &gt; n d s(n)=\sum _{d\mid n,~d&gt;n} d denote the sum of the proper divisors of n n . The second-named author proved that ω ( s ( n ) ) \omega (s(n)) has normal order log ⁡ log ⁡ n \log \log {n} , the analogue for s s -values of a classical result of Hardy and Ramanujan [The normal number of prime factors of a number n [Quart. J. Math. 48 (1917), 76–92], AMS Chelsea Publ., Providence, RI, 2000, pp. 262–275]. We establish the corresponding Erdős–Kac theorem: ω ( s ( n ) ) \omega (s(n)) is asymptotically normally distributed with mean and variance log ⁡ log ⁡ n \log \log {n} . The same method applies with s ( n ) s(n) replaced by any of several other unconventional arithmetic functions, such as β ( n ) ≔ ∑ p ∣ n p \beta (n)≔\sum _{p\mid n} p , n − φ ( n ) n-\varphi (n) , and n + τ ( n ) n+\tau (n) ( τ \tau being the divisor function).

The adjacency spectrum and metric dimension of an induced subgraph of comaximal graph of ℤn

Let R be a commutative ring with unity, and let [Formula: see text] denote the comaximal graph of R. The comaximal graph [Formula: see text] has vertex set as R, and any two distinct vertices x, y of [Formula: see text] are adjacent if [Formula: see text]. Let [Formula: see text] denote the induced subgraph of [Formula: see text] on the set of all nonzero non-unit elements of R, and any two distinct vertices x, y of [Formula: see text] are adjacent if [Formula: see text]. In this paper, we study the graphical structure as well the adjacency spectrum of [Formula: see text], where [Formula: see text] is a non-prime positive integer, and [Formula: see text] is the ring of integers modulo n. We show that for a given non-prime positive integer n with D number of positive proper divisors, the eigenvalues of [Formula: see text] are [Formula: see text] with multiplicity [Formula: see text], and remaining eigenvalues are contained in the spectrum of a symmetric [Formula: see text] matrix. We further calculate the rank and nullity of [Formula: see text]. We also determine all the eigenvalues of [Formula: see text] whenever [Formula: see text] is a bipartite graph. Finally, apart from determining certain structural properties of [Formula: see text], we conclude the paper by determining the metric dimension of [Formula: see text].

Period Collapse in Characteristic Quasi-Polynomials of Hyperplane Arrangements

Abstract Given an integral hyperplane arrangement, Kamiya–Takemura–Terao [11, 12] introduced the notion of characteristic quasi-polynomial, which enumerates the cardinality of the complement of the arrangement modulo a positive integer. The most popular candidate for periods of the characteristic quasi-polynomials is the lcm period. In this paper, we initiate a study of period collapse in characteristic quasi-polynomials stemming from the concept of period collapse in the theory of Ehrhart quasi-polynomials. We say that period collapse occurs in a characteristic quasi-polynomial when the minimum period is strictly less than the lcm period. Our first main result is that in the non-central case, with regard to period collapse anything is possible: period collapse occurs in any dimension $\ge 1$, occurs for any value $\ge 2$ of the lcm period, and the minimum period when it is not the lcm period can be any proper divisor of the lcm period. Our second main result states that in the central case, however, no period collapse is possible in any dimension, that is, the lcm period is always the minimum period.

Powerfree sums of proper divisors

Let $s(n):= \sum _{d\,|\, n,\,d \lt n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $k\ge 2$, the equivalence \[ \text {$n$ is $k$th powerfree} \iff \text {$s(n)$ is $k$th powerfree} \] holds almost al

On the adjacency spectrum of zero divisor graph of ring ℤn

The zero divisor graph [Formula: see text] of a commutative ring [Formula: see text] with unity is a simple undirected graph whose vertices are all nonzero zero divisors of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study the graphical structure and the adjacency spectrum of the zero divisor graph of ring [Formula: see text]. For any non-prime positive integer [Formula: see text] with [Formula: see text] number of proper divisors, we show that the adjacency spectrum of [Formula: see text] consists of the eigenvalues of a symmetric matrix [Formula: see text] of size [Formula: see text], and at the most [Formula: see text] and [Formula: see text]. Also, we find the exact multiplicity of the eigenvalue [Formula: see text] and show that all eigenvalues of [Formula: see text] are nonzero, by determining the rank and nullity of the adjacency matrix of [Formula: see text]. We find the values of [Formula: see text] for which the adjacency spectrum of [Formula: see text] contains only nonzero eigenvalues. Finally, by computing the characteristic polynomial of the matrix [Formula: see text], we determine the characteristic polynomial of [Formula: see text] whenever [Formula: see text] is a prime power.

Divisor sums representable as the sum of two squares

Let $s(n)$ denote the sum of the proper divisors of the natural number $n$. We show that the number of $n \leq x$ such that $s(n)$ is a sum of two squares has order of magnitude $x/\sqrt{\log x}$, which agrees with the count of $n \leq x$ which are a sum of two squares. Our result confirms a special case of a conjecture of Erd{\H o}s, Granville, Pomerance and Spiro, who in a 1990 paper asserted that if $\mathcal{A} \subset \mathbb{N}$ has asymptotic density zero (e.g. if $\mathcal{A}$ is the set of $n \leq x$ which are a sum of two squares), then $s^{-1}(\mathcal{A})$ also has asymptotic density zero.

Some New Notes on Mersenne Primes and Perfect Numbers

&lt;p&gt;Mersenne primes are specific type of prime numbers that can be derived using the formula &lt;img title="\large M_p=2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;amp;space;M_p=2^{p}-1" alt="" /&gt;, where &lt;img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;amp;space;p" alt="" /&gt; is a prime number. A perfect number is a positive integer of the form &lt;img title="\large P(p)=2^{p-1}(2^{p}-1)" src="https://latex.codecogs.com/gif.latex?\large&amp;amp;space;P(p)=2^{p-1}(2^{p}-1)" alt="" /&gt; where &lt;img title="\large 2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;amp;space;2^{p}-1" alt="" /&gt; is prime and &lt;img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;amp;space;p" alt="" /&gt; is a Mersenne prime, and that can be written as the sum of its proper divisor, that is, a number that is half the sum of all of its positive divisor. In this note, some concepts relating to Mersenne primes and perfect numbers were revisited. Further, Mersenne primes and perfect numbers were evaluated using triangular numbers. This note also discussed how to partition perfect numbers into odd cubes for odd prime &lt;img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;amp;space;p" alt="" /&gt;. Also, the formula that partition perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts. The results of this study is useful to better understand the mathematical structure of Mersenne primes and perfect numbers.&lt;/p&gt;

Thanks for Twenty-Four

The number 24 belongs to a variety of integer sequences: 24 is the factorial of 4; the sum of its proper divisors, 1 + 2 + 3 + 4 + 6 + 8 + 12 = 36, exceeds 24 making it abundant; the sum of digits ...

On near-perfect numbers

The study of perfect numbers (numbers which equal the sum of their proper divisors) goes back to antiquity, and is responsible for some of the oldest and most popular conjectures in number theory. We investigate a generalization introduced by Pollack and

Orders generated by character values

Let $K:=\mathbb{Q}(G)$ be the number field generated by the complex character values of a finite group $G$. Let $\mathbb{Z}_K$ be the ring of integers of $K$. In this paper we investigate the suborder $\mathbb{Z}[G]$ of $\mathbb{Z}_K$ generated by the character values of $G$. We prove that every prime divisor of the order of the finite abelian group $\mathbb{Z}_K/\mathbb{Z}[G]$ divides $|G|$. Moreover, if $G$ is nilpotent, we show that the exponent of $\mathbb{Z}_K/\mathbb{Z}[G]$ is a proper divisor of $|G|$ unless $G=1$. We conjecture that this holds for arbitrary finite groups $G$.

Laplacian eigenvalues of the zero divisor graph of the ring [formula omitted

We study the Laplacian eigenvalues of the zero divisor graph Γ(Zn) of the ring Zn and prove that Γ(Zpt) is Laplacian integral for every prime p and positive integer t≥2. We also prove that the Laplacian spectral radius and the algebraic connectivity of Γ(Zn) for most of the values of n are, respectively, the largest and the second smallest eigenvalues of the vertex weighted Laplacian matrix of a graph which is defined on the set of proper divisors of n. The values of n for which algebraic connectivity and vertex connectivity of Γ(Zn) coincide are also characterized.

Integral circulant graphs with four distinct eigenvalues

Let [Formula: see text], the finite cyclic group of order [Formula: see text]. Assume that [Formula: see text] and [Formula: see text]. The circulant graph [Formula: see text] is the undirected graph having the vertex set [Formula: see text] and edge set [Formula: see text]. Let [Formula: see text] be a set of positive, proper divisors of the integer [Formula: see text]. In this paper, by using [Formula: see text] we characterize certain connected integral circulant graphs with four distinct eigenvalues.

Mystery of ‘Perfect Numbers’ Resolved – Perfect Number Is Always Even and Predictable

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, excluding the number itself. In other words, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n. To explain in practical terms, we elaborate first few Perfect Numbers. It may be noted that ‘Perfect Numbers’ are sparse are thinly dispersed. Starting from 3rd Century BC, mathematicians are working on Perfect Numbers. Till April 2018, i.e. during last 2300 years active research, researchers could recognize only 50 perfect numbers. There are 2 perfect numbers in first 100 and 4 in first million. Absolute distance between two perfect numbers increase exponentially as you go higher to the next perfect number . One can find at least one perfect number till 4 digit numbers, and then it becomes a real rarity. Subsequent perfect numbers appears at 8, 10, 12 and 19 digits. 15th perfect number has 770 digits while 16th have 1327 digits. 25th perfect number has 13066 digits. 50th perfect number has 46,498,850 digits. We found that perfect number is always predictable by using formula 1 (p) 0 (p-1) where 1 and 0 are binary digits and p = count of binary digit. We also argue that if any binary number 1...(p) 0 (p-1) if perfect number, will always an even number. We also observed that first known 50 perfect number ends with 6 or 28 as last one or two digits. Therefore a perfect number is always predictable and even.