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Consecutive generalized r-free integers in Beatty sequences
(2023-03-01)
Srisopha, Sunanta
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Srichan, Teerapat
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Mavecha, Sukrawan
In this paper, we use the technique of Tangsupphathawat et al. to generalize a previous result on the distribution of consecutive square-free integers in Beatty sequences. Moreover, the distribution of consecutive semi r-free integers in Beatty sequences is also studied.
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On r-free integers in Beatty sequences
(2022-07-01)
Kim, Veasna
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Srichan, Teerapat
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Mavecha, Sukrawan
Let r≥ 2 be a fixed integer. A positive integer n is called r-free if in the canonical representation of n into prime powers each exponent is < r. The integer 1 is considered to be r-free. In this paper, we consider Q<inf>r</inf>(x; α, β) , which is the number of r-free integers of Beatty sequence ⌊ αn+ β⌋ , 1 ≤ n≤ x, for α> 1 irrational and with bounded partial quotients, β∈ [0 , α). We prove that, as x→ ∞Qr(x;α,β)=xζ(r)+O(x(r+1)/2rlog3x),which improves Victorovich’s result in the case of square-free integers. Moreover, we also prove there exist infinitely many consecutive square-free numbers of the forms ⌊ αn+ β⌋ , ⌊ αn+ β⌋ + 1 , which improves Dimitrov’s result in 2019.
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A divided-difference characterization of polynomials over finite fields of characteristic two
(2022-04-01)
Kim, Veasna
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Laohakosol, Vichian
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Mavecha, Sukrawan
For a finite field F of characteristic 2, an integer n≥ 3 , and for a function f: F→ F, if there is a function h: F→ F such that the divided difference f[x<inf>1</inf>, ⋯ , x<inf>n</inf>] on any n distinct elements of F satisfies f[x<inf>1</inf>, … , x<inf>n</inf>] = h(x<inf>1</inf>+ ⋯ + x<inf>n</inf>) , then f is a polynomial of degree at most n over F. This makes earlier work of Davies and Rousseau complete.
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Odd/Even r-Free Numbers
(2022-01-01)
Srisopha, Sunanta
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Srichan, Teerapat
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Mavecha, Sukrawan
In this paper we use an elementary method to give an asymptotical ratio of odd to even r-free numbers and show that it is asymptotically 2r: 2r − 2.
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A class of continuous solutions of a fourth order polynomiallike iterative equation
(2021-07-01)
Thangroongvongthana, Supharerg
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Laohakosol, Vichian
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Mavecha, Sukrawan
Using the so-called characteristic method, continuous solutions of the fourth order polynomial-like iterative equation 4323210 f4(x) + a3f3(x) + a2f2(x) + a1f(x) + a0x=0 were determined subject to certain natural conditions on its characteristic roots. The result so obtained complements earlier work in the cases of second and third order equations.
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Integer factorization and finite Fourier series expansion
(2021-04-01)
Yangklan, Pussadee
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Laohakosol, Vichian
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Mavecha, Sukrawan
Given two integers a and k > 0, the number of factorizations of a (mod k) is the number of ordered pairs (s, t) ∈ {0, 1,.. , k − 1}2 satisfying s · t ≡ a (mod k). This number is known to be expressible by a formula involving the greatest common divisor function. Motivated by such a formula, we derive several formulae counting the number of factorizations of a (mod k) subject to certain other natural restrictions. Some of these formulae are obtained as consequences of finite Fourier series expansions of the greatest common divisor function, whereas some are shown to be closely connected with the notion of unitary divisors.
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Formulae of the frobenius number in relatively prime three lucas numbers
(2020-01-01)
Bokaew, Ratchanok
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Yuttanan, Boonrod
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Mavecha, Sukrawan
In this paper, we find the explicit formulae of the Frobenius number for numerical semigroups generated by relatively prime three Lucas numbers L<inf>i</inf>, L<inf>i+2</inf> and L<inf>i+l</inf> for given integers i ≥ 3,l≥ 4.
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Linear dependence of four types of arithmetic functions
(2019-11-01)
Ponpetch, Kanet
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Laohakosol, Vichian
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Mavecha, Sukrawan
General criteria were derived for the linear dependence of arithmetic functions over the complex field as well as several other criteria for arithmetic functions that were solutions of additive, multiplicative, exponential, and logarithmic equations. A number of examples were worked out in order to compare the results with the existing ones.
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Independence of additive, multiplicative, exponential and logarithmic functions
(2019-10-01)
Ponpetch, Kanet
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Laohakosol, Vichian
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Mavecha, Sukrawan
The independence of four types of functions, namely, additive, multiplicative, exponential and logarithmic functions defined over two types of domains, namely, discrete and non-discrete sets are investigated.
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A system of functional equations satisfied by components of a quadratic function and its stability
(2019-10-01)
Ponpetch, Kanet
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Laohakosol, Vichian
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Mavecha, Sukrawan
A system of functional equations satisfied by the components of a quadratic function is derived via their corresponding circulant matrix. Given such a system of functional equations, general solutions are determined and a stability result for such a system is established.
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