On Two Problems Related to Divisibility Properties of z(n)

The order of appearance (in the Fibonacci sequence) function z:Z≥1→Z≥1 is an arithmetic function defined for a positive integer n as z(n)=min{k≥1:Fk≡0(modn)}. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z(n)=z(n2), where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z(n)=z(n+1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that limsupn→∞(z(n+1)−z(n))/(logn)2−ϵ=∞, for all ϵ∈(0,2).

Analytical properties of type 2 degenerate poly-Bernoulli polynomials associated with their applications

Recently, Kim et al. (Adv. Differ. Equ. 2020:168, 2020) considered the poly-Bernoulli numbers and polynomials resulting from the moderated version of degenerate polyexponential functions. In this paper, we investigate the degenerate type 2 poly-Bernoulli numbers and polynomials which are derived from the moderated version of degenerate polyexponential functions. Our degenerate type 2 degenerate poly-Bernoulli numbers and polynomials are different from those of Kim et al. (Adv. Differ. Equ. 2020:168, 2020) and Kim and Kim (Russ. J. Math. Phys. 26(1):40–49, 2019). Utilizing the properties of moderated degenerate poly-exponential function, we explore some properties of our type 2 degenerate poly-Bernoulli numbers and polynomials. From our investigation, we derive some explicit expressions for type 2 degenerate poly-Bernoulli numbers and polynomials. In addition, we also scrutinize type 2 degenerate unipoly-Bernoulli polynomials related to an arithmetic function and investigate some identities for those polynomials. In particular, we consider certain new explicit expressions and relations of type 2 degenerate unipoly-Bernoulli polynomials and numbers related to special numbers and polynomials. Further, some related beautiful zeros and graphical representations are displayed with the help of Mathematica.

A short remark on an arithmetic function

An explicit form of A. Shannon’s arithmetic function δ is given. A possible application of it is discussed for representation of the well-known arithmetic functions ω and Kronecker’s delta-function δ_{m,s}.

On Vandiver’s arithmetical function – I

We study certain properties of Vandiver’s arithmetic function V(n) = \prod_{d|n} (d+1).

The Properties of the Arithmetic Function as the Consecutive Sum of Digits in Natural Numbers

In this paper defines the consecutive sum of the digits of a natural number, so far as it becomes less than ten, as an arithmetic function called and then introduces some important properties of this function by proving a few theorems in a way that they can be used as a powerful tool in many cases. As an instance, by introducing a test called test, it has been shown that we are able to examine many algebraic equalities in the form of in which and are arithmetic functions and to easily study many of the algebraic and diophantine equations in the domain of whole numbers. The importance of test for algebraic equalities can be considered equivalent to dimensional equation in physics relations and formulas. Additionally, this arithmetic function can also be useful in factorizing the composite odd numbers.

Using Threshold DNA Strand Displacement to Construct a Half Adder

DNA strand displacement has the advantages of product predictability, programmability and threshold, and it is often used to build DNA-based logic operation systems. In this paper, we use DNA strand displacement to have different reaction priorities in different length ranges of the toehold domain to form the effect of the threshold and construct the logical AND gate and XOR gate. Logical operations use single-stranded DNA as the input signal, and the brightness of the fluorescence is used to measure the results. Then, the logic AND gate and XOR gate are used as basic logic units to form a half adder in parallel. Finally, we use Visual DSD to simulate and analyze the logic AND gate, XOR gate and half adder. The simulation results show that the logic gates constructed in this paper have good theoretical feasibility and effectiveness. This work provides a potential design idea for DNA-based arithmetic function operations and more advanced logic operations.

A New Family of Degenerate Poly-Genocchi Polynomials with Its Certain Properties

In this paper, we introduce a new type of degenerate Genocchi polynomials and numbers, which are called degenerate poly-Genocchi polynomials and numbers, by using the degenerate polylogarithm function, and we derive several properties of these polynomials systematically. Then, we also consider the degenerate unipoly-Genocchi polynomials attached to an arithmetic function, by using the degenerate polylogarithm function, and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

On the Sandor-Smarandache Function

In the current literature, a new Smarandache-type arithmetic function, involving binomial coefficients, has been proposed by Sandor. The new function, denoted by SS(n), is named the Sandor-Smarandache function. It has been found that, like many Smarandache-type functions, SS(n) is not multiplicative. Sandor found SS(n) when n (≥3) is an odd integer. Since then, the determination of SS(n) for even n remains a challenging problem. It has been shown that the function has a simple form even when n is even and not divisible by 3. This paper finds SS(n) in some particular cases of n, and finds an upper bound of SS(n) for some special forms of n. Some equations involving the Sandor-Smarandache function and pseudo-Smarandache function have been studied. A list of values of SS(n) for n = 1(1)480, calculated on a computer, is appended at the end of the paper.

Stieltjes constants of L-functions in the extended Selberg class

Let f be an arithmetic function and let $${\mathcal {S}}^\#$$ S # denote the extended Selberg class. We denote by $${\mathcal {L}}(s) = \sum _{n = 1}^{\infty }\frac{f(n)}{n^s}$$ L ( s ) = ∑ n = 1 ∞ f ( n ) n s the Dirichlet series attached to f. The Laurent–Stieltjes constants of $${\mathcal {L}}(s)$$ L ( s ) , which belongs to $${\mathcal {S}}^\#$$ S # , are the coefficients of the Laurent expansion of $${\mathcal {L}}$$ L at its pole $$s=1$$ s = 1 . In this paper, we give an upper bound of these constants, which is a generalization of many known results.