New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))
-
Djamel Bellaouar
,
Alain Togbé
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.
This research work is supported by the General Direction of Scientific Research and Technological Development (DGRSDT)-Algeria.
Acknowledgement
This paper started when the second author was visiting Max-Planck--Institute für Mathematik. He thanks the institution for the great working environment, the hospitality, and the support.
Communicated by István Gaál
References
[1] Amroune, Z.—Bellaouar, D.—Boudaoud, A.: A class of solutions of the equation d(n2) = d(φ(n)), Notes Number Theory Discrete Math. 29(2) (2023), 284–309.10.7546/nntdm.2023.29.2.284-309Search in Google Scholar
[2] Bellaouar, D.—Boudaoud, A.—Özer, Ö.: On a sequence formed by iterating a divisor operator, Czechoslovak Math. J. 69 (2019), 1177–1196.10.21136/CMJ.2019.0133-18Search in Google Scholar
[3] Bellaouar, D.—Boudaoud, A.—Jakimczuk, R.: Notes on the equation d(n) = d(φ (n)) and related inequalities, Math. Slovaca 73(3) (2023), 613–632.10.1515/ms-2023-0045Search in Google Scholar
[4] De Koninck, J. M.—Mercier, A.: 1001 Problems in Classical Number Theory, American Mathematical Society, Providence, 2007.Search in Google Scholar
[5] Duber, H.: Large Sophie Germain primes, Math. Comp. 65(213) (1996), 393–396.10.1090/S0025-5718-96-00670-9Search in Google Scholar
[6] Nathanson, M. B.: Elementary Methods in Number Theory, Springer-Verlag, New York, 2000.Search in Google Scholar
[7] Sándor, J.: Geometric Theorems, Diophantine Equations, and Arithmetic Functions, American Research Press. Rehoboth, 2002.Search in Google Scholar
[8] Sándor, J.—Bhattacharjee, S.: On certain equations and inequalities involving the arithmetical functions φ (n) and d(n), Notes Number Theory Discrete Math. 28(2) (2022), 376–379.10.7546/nntdm.2022.28.2.376-379Search in Google Scholar
[9] Sándor, J.: On certain equations and inequalities involving the arithmetical functions φ (n) and d(n) -II, Notes Number Theory Discrete Math. 29(1) (2023), 130–136.10.7546/nntdm.2023.29.1.130-136Search in Google Scholar
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Articles in the same Issue
- On monoids of endomorphisms of a cycle graph
- The influence of separating cycles in drawings of K5 ∖ e in the join product with paths and cycles
- Completely hereditarily atomic OMLS
- New notes on the equation d(n) = d(φ(n)) and the inequality d(n) > d(φ(n))
- On the monogenity and Galois group of certain classes of polynomials
- Other fundamental systems of solutions of the differential equation (D2 – 2αD + α2 + β2)ny = 0, β ≠ 0
- Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces
- Hermite–Hadamard inequalities for Riemann–Liouville fractional integrals
- Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation
- The discrete fractional Karamata theorem and its applications
- Complex Generalized Stancu-Schurer Operators
- New oscillatory criteria for third-order differential equations with mixed argument
- Cauchy problem for a loaded hyperbolic equation with the Bessel operator
- Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p0)-condition
- On the rate of convergence for the q-Durrmeyer polynomials in complex domains
- Some fixed point theorems in generalized parametric metric spaces and applications to ordinary differential equations
- On the relation of Kannan contraction and Banach contraction
- Extropy and statistical features of dual generalized order statistics’ concomitants arising from the Sarmanov family
- Residual Inaccuracy Extropy and its properties
- Archimedean ℓ-groups with strong unit: cozero-sets and coincidence of types of ideals
