Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
-
Djamel Bellaouar
,
Abdelmadjid Boudaoud
ABSTRACT
Let d(n) denotes the number of positive integers dividing the positive integer n, and let φ(n) denotes Euler’s function representing the number of numbers less than and prime to n. In this paper, we present some notes on the equation d(n) = d(φ(n)). Several results on the related inequalities are also obtained.
Funding statement: This research work is supported by the General Direction of Scientific Research and Technological Development (DGRSDT)-Algeria.
Acknowledgement
We thank the referee for very helpful and detailed comments which improved the quality of this paper. In particular, the proof of assertion (v) in Corollary 2.2.1, the proof of Proposition 3.2 and Question (4) in Section 5 came from him. The authors would like to thank Professors Karel Hrbáček and A. Satyanarayana Reddy for the careful reading of the manuscript and helpful remarks.
REFERENCES
[1] Alford, W. R.—Granville, A.—Pomerance, C.: There are infinitely many Carmichael numbers. Ann. of Math. 139 (1994), 703–722.10.2307/2118576Suche in Google Scholar
[2] Bellaouar, D.—Boudaoud, A.—Özer, Ö.: On a sequence formed by iterating a divisor operator, Czechoslovak Math. J. 69(144) (2019), 1177–1196.10.21136/CMJ.2019.0133-18Suche in Google Scholar
[3] De Koninck, J. M.—Mercier, A.: 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, 2007.Suche in Google Scholar
[4] Dubner, H.: Large Sophie Germain primes, Math. Comp. 65(213) (1996), 393–396.10.1090/S0025-5718-96-00670-9Suche in Google Scholar
[5] Guy, R. K.: Unsolved Problems in Number Theory, 2. ed., Springer-Verlag, New York, 1994.10.1007/978-1-4899-3585-4Suche in Google Scholar
[6] Heath-Brown, D. R.: The divisor function at consecutive integers, Mathematika 31 (1984), 141–149.10.1112/S0025579300010743Suche in Google Scholar
[7] Iannucci, D. E.: On the equation σ(n) = n + φ(n), J. Integer Seq. 20 (2017), Art. ID 17.6.2.Suche in Google Scholar
[8] Liu, F.: On the Sophie Germain prime conjecture, WSEAS Transactions on Mathematics 10(12) (2011), 421–430.Suche in Google Scholar
[9] Luca, F.: Equations involving arithmetic functions of factorials, Divulgaciones Matemáticas 8 (2000), 15–23.Suche in Google Scholar
[10] Nicol, C. A.: Some Diophantine equations involving arithmetic functions, J. Math. Anal. Appl. 15 (1966), 154–161.10.1016/0022-247X(66)90148-XSuche in Google Scholar
[11] Pinch, R. G.: The Carmichael numbers up to 1015, Math. Comp. 61(203) (1993), 381–91.10.1090/S0025-5718-1993-1202611-7Suche in Google Scholar
[12] Ramanujan, S.: Highly composite numbers, Proc. Lond. Math. Soc. 1 (1915), 347–409.10.1112/plms/s2_14.1.347Suche in Google Scholar
[13] Sándor, J.: Geometric Theorems, Diophantine Equations, and Arithmetic Functions, American Research Press, Rehoboth, 2002.Suche in Google Scholar
[14] Tattersall, J. J.: Number Theory in Nine Chapters, Second Edition, Cambridge University Press, 2005.10.1017/CBO9780511756344Suche in Google Scholar
[15] Wigert, S.: Sur L’ordre de Grandeur du Nombre des Diviseurs d’un Entier, Almqvist & Wiksell, Uppsala, 1907.Suche in Google Scholar
- 1
Note that if d(m) > 2k–1 and t ≥ 1, then (3.3) is not true.
© 2023 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables
Artikel in diesem Heft
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables
