Positive Divisors Research Articles

New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))

Abstract Let d(n) and φ (n) denote the number of positive divisors of n and the Euler’s phi function of n, respectively. In this paper, we prove various (conditional and unconditional) results about the solvability of the Diophantine equation d(n) = d(φ(n)) and a related inequality. For further research, we present some open problems.

Euler’s partition function in terms of 2-adic valuation

The 2-adic valuation of an integer n is the exponent of the highest power of 2 that divides n and is denoted by ν2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _2(n)$$\\end{document}. In this paper, we prove that Euler’s partition function p(n) can be expressed in terms of ν2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _2(n)$$\\end{document}. Our approach allows us to express the sum of positive divisors of n in terms of ν2(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _2(n)$$\\end{document}. We introduce the notion of 2-adic color partition and provide a new combinatorial interpretation for Euler partition function p(n). Connections between partitions and the game of m-Modular Nim with two heaps are presented in this context.

On unitary Zumkeller numbers

It is well known that if n is a Zumkeller number, then the positive divisors of n can be partitioned into two disjoint subsets of equal sum. Similarly for unitary Zumkeller number n, the unitary divisors of n can be partitioned into two disjoint subsets of equal sum. In this article, we have derived some results related to unitary Zumkeller number, unitary half-Zumkeller number and also presented some numerical examples.

Explicit evaluation of triple convolution sums of the divisor functions

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums [Formula: see text] for odd integers [Formula: see text] and [Formula: see text], where [Formula: see text] is the sum of the [Formula: see text]th powers of the positive divisors of [Formula: see text]. We consider four cases, namely (i) [Formula: see text], (ii) [Formula: see text]; [Formula: see text] (iii) [Formula: see text]; [Formula: see text] and (iv) [Formula: see text], and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer [Formula: see text] by certain positive definite quadratic forms. The existing formulas for [Formula: see text] (in [20]), [Formula: see text] (in [7]), [Formula: see text] (in [35]), [Formula: see text], [Formula: see text] (in [30]) and [Formula: see text] (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.

Arithmetic functions at factorial arguments

For various arithmetic functions f : N ? R, the behavior of f(n!) and that of ? n?N f(n!) can be intriguing. For instance, for some functions f, we have f(n!) = ? k?n f(k), for others, we have f(n!) = ? p?n f(p) (where the sum runs over all the primes p ? n). Also, for some f, their minimum order coincides with lim n? ?f(n!), for others, it is their maximum order that does so. Here, we elucidate such phenomena and more generally, we embark on a study of f(n!) and of ? n?N f(n!) for a wide variety of arithmetical functions f. In particular, letting d(n) and ?(n) stand respectively for the number of positive divisors of n and the sum of the positive divisors of n, we obtain new accurate asymptotic expansions for d(n!) and ? (n!). Furthermore, setting ? 1(n) := max{d | n : d ? ?n} and observing that no one has yet obtained an asymptotic value for ? n?N ? 1(n) as N ? ?, we show how one can obtain the asymptotic value of ? n?N ?1(n!).

On a modification of $\underline{Set}(n)$

A modification of the set $\underline{\rm Set}(n)$ for a fixed natural number $n$ is introduced in the form: $\underline{\rm Set}(n, f)$, where $f$ is an arithmetic function. The sets $\underline{\rm Set}(n,\varphi), \underline{\rm Set}(n,\psi), \underline{\rm Set}(n,\sigma)$ are discussed, where $\varphi, \psi$ and $\sigma$ are Euler's function, Dedekind's function and the sum of the positive divisors of $n$, respectively.

R-primitive k-normal elements in arithmetic progressions over finite fields

Let be a finite field with qn elements. For a positive divisor r of , the element is called r-primitive if its multiplicative order is . Also, for a non-negative integer k, the element is k-normal over if in has degree k. In this paper we discuss the existence of elements in arithmetic progressions with being ri -primitive and at least one of the elements in the arithmetic progression being k-normal over . We obtain asymptotic results for general and concrete results when for .

Generalized Euler's \( \Phi_w \)-function and the divisor sum \(T_{k_w} \)-function of edge weighted graphs

In this work, generalized Euler's \(\Phi_w \)-function of edge weighted graphs is defined which consists of the sum of the Euler's \(\varphi\)-function of the weight of edges of a graph and we denote it by \(\Phi_w(G)\) and the general form of Euler's \(\Phi_w\)-function of some standard edge weighted graphs is determined. Also, we define the divisor sum \(T_{k_w}\)-function \(T_{k_w}(G)\) of the graph \(G\), which is counting the sum of the sum of the positive divisor \(\sigma_k \)-function for the weighted of edges of a graph \(G\). It is determined a relation between generalized Euler's \(\Phi_w\)-function and generalized divisor sum \(T_{k_w}\)-function of edge weighted graphs.

NOTE ON THE TWO CELEBRITY NUMBERS, PERFECT NUMBERS AND TRIANGULAR NUMBERS

Mathematicians have been fascinated for centuries by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). Such numbers are called perfect numbers. Thus a positive integer is called a perfect number if it is equal to the sum of its proper positive divisors. The search for perfect numbers began in ancient times. The four perfect numbers 6, 28, 496, and 8128 seem to have been known from ancient times. In this paper, we will investigate some important properties of perfect numbers. We give easy and simple proofs of theorems using finite series. We give our own alternative proof of the well-known Euclid’s Theorem
 (Theorem I). We will also prove some important theorems which play key roles in the mathematical theory of perfect numbers.

Complete (k,r)-Caps From Orbits In PG(3,11)

The purpose of this article is to partition PG(3,11) into orbits. These orbits are studied from the view of caps using the subgroups of PGL(4,11) which are determined by nontrivial positive divisors of the order of PG(3,11). The τ_i-distribution and c_i-distribution are also founded for each cap.

A NOTE ON A CLASSICAL CONNECTION BETWEEN PARTITIONS AND DIVISORS

In this note, we consider the number of k’s in all the partitions of n in order to provide a new proof of a classical identity involving Euler’s partition function p(n) and the sum of the positive divisors function a(n). New relations connecting classical functions of multiplicative number theory with the partition function p(n) from additive number theory are introduced in this context. The fascinating feature of these relations is their common nature. A new identity for the number of 1’s in all the partitions of n is derived in this context.

On the Ramanujan-type congruences modulo 8 for the overpartitions into odd parts

Let denote the number of overpartitions of n into odd parts. In this paper, we provide a complete characterization of Ramanujan-type congruences modulo 8 for the overpartition function considering the number of odd positive divisors of and the relations of the form with and

Construction of Complete (k;r)-Arcs from Orbits in PG(3,11)

The aim of this research is to partition into orbits using the subgroups of which are determined by the nontrivial positive divisors of the order of . These orbits were also studied from the perspective of arcs by finding complete and incomplete arcs.

On Bounds of Non-Deficient Numbers

Objectives: To improve the upper bounds of a quasi perfect number and give an important result on its divisibility with primes. Methods: A positive integer n is quasi perfect if s (n) >2n + 1, where s (n) denotes the sum of the positive divisors of n. However, the existence of a quasi perfect number, which is a Non-Deficient number, is still an open problem. We use R(n), the sum of the reciprocals of distinct primes dividing the quasi perfect number, to derive lemmas and improve the bounds obtained by earlier authors. Findings: We improve the upper bounds for R(n), when n is quasi perfect with gcd (15, n) = 3 or gcd (15, n) = 5. As a consequence, we establish that a quasi perfect number, if exists, is divisible by both 3 and 5 or by none of them. Novelty: The unique method of using R(n) also resulted in finding an important result that 3, 5 and 7 cannot divide any quasi perfect number. Mathematics Subject Classification: 11A05, 11A25. Keywords: non-deficient number; quasi perfect number; sum of the divisor; sum of the reciprocal; bounds of perfect number; number of divisors

Families of Ramanujan-Type Congruences Modulo 4 for the Number of Divisors

In this paper, we explore Ramanujan-type congruences modulo 4 for the function σ0(n), counting the positive divisors of n. We consider relations of the form σ08(αn+β)+r≡0(mod4), with (α,β)∈N2 and r∈{1,3,5,7}. In this context, some conjectures are made and some Ramanujan-type congruences involving overpartitions are obtained.

Near-Zumkeller numbers

A positive integer n is called a Zumkeller number if the set of all the positive divisors of n can be partitioned into two disjoint subsets, each summing to σ(n)/2. In this paper, Generalizing further, near-Zumkeller numbers and k-near-Zumkeller numbers are defined and also some results concerning these numbers are established. Relations of these numbers with practical numbers are also studied in this paper.

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].

Arithmetic convolution sums derived from eta quotients related to divisors of 6

Abstract The aim of this paper is to find arithmetic convolution sums of some restricted divisor functions. When divisors of a certain natural number satisfy a suitable condition for modulo 12, those restricted divisor functions are expressed by the coefficients of certain eta quotients. The coefficients of eta quotients are expressed by the sine function and cosine function, and this fact is used to derive formulas for the convolution sums of restricted divisor functions and of the number of divisors. In the sine function used to find the coefficients of eta quotients, the result is obtained by utilizing a feature with symmetry between the divisor and the corresponding divisor. Let N , r N,r be positive integers and d d be a positive divisor of N N . Let e r ( N ; 12 ) {e}_{r}\left(N;\hspace{0.33em}12) denote the difference between the number of 2 N d − d \frac{2N}{d}-d congruent to r r modulo 12 and the number of those congruent to − r -r modulo 12. The main results of this article are to find the arithmetic convolution identities for ∑ a 1 + ⋯ + a j = N ( ∏ i = 1 j e ˆ ( a i ) ) {\sum }_{{a}_{1}+\cdots +{a}_{j}=N}({\prod }_{i=1}^{j}\hat{e}\left({a}_{i})) with e ˆ ( a i ) = e 1 ( a i ; 12 ) + 2 e 3 ( a i ; 12 ) + e 5 ( a i ; 12 ) \hat{e}\left({a}_{i})={e}_{1}\left({a}_{i};\hspace{0.33em}12)+2{e}_{3}\left({a}_{i};\hspace{0.33em}12)+{e}_{5}\left({a}_{i};\hspace{0.33em}12) and j = 1 , 2 , 3 , 4 j=1,2,3,4 . All results are obtained using elementary number theory and modular form theory.

Refinement of the upper bound of the radius of a circular integral points set

Let n be a product of the prime numbers whose positive integer powers are of the form a2+Db2 where D> 4 is a square-free number and a, b are positive integers. For n≤ 3072, we obtained a refinement of the upper bound of the radius of a circular points set which was previously given in Tables 1 and 2 of Ganbileg's paper. In order to prove this, we showed that there are points on the circle with the radius R=n √D/2D such that mutual distances between these points are all integers. Consequently, if n is a product of the prime numbers whose squares are of the form a2+Db2, then we showed that there are τ(n) points on the circle with the radius R=n √D/2D such that mutual distances between these points are all integer numbers, where τ(n) is the number of all positive divisors of n.

Some new semiring structures

In this paper, we introduce some new semiring structures by considering the prime factors, sum of divisors and the number of relatively prime positive divisors of a pair of non-negative integers and solve a variety of system of linear equations over one of these semirings. We then discuss the properties of these structures and how they compare with classical algebra. The paper is concluded by providing a key-exchange scheme using one of the newly discovered semirings.