Two arithmetic operations are introduced and some of their properties are studied. It is proved that they can be operations of semi-groups, but not of monoids. It is shown that their reverse operations are not one-valued. Some connections between the new operations and the well-known arithmetic functions φ, Ψ, σ are shown.

This study concerns some properties of the generalized nonhomogeneous Horadam model and its application to four types of known sequences of numbers. Regarding the four sequences of this model, we provide results on combinatorial and linear expressions, and also on the analytical representation of Binet. Our approach is based on results we have developed regarding the general setting of nonhomogeneous linear recursive sequences, especially the fundamental sequence related to their homogeneous part.

In this paper, we present explicit analytical expressions for certain families of log-sine definite integrals: \int_0^{2\pi} x^{m}\{ \ln(2\sin\frac{x}{2})\}^{n}dx \ (m\in\mathbb{N}_{0}, n\in\mathbb{N}), expressed in terms of multiple hypergeometric functions of the Kampé de Fériet with arguments ±1 and the Riemann zeta functions. As applications, we establish several mixed summation formulas (79), (81) and (83) involving the generalized hypergeometric functions _3F_2(1), _5F_4(1) and _7F_6(1). Moreover, a collection of possibly new summation formulas (42), (52), (54), (56), (58), (62), (64), (66), (70), (72), (74) and (76) for multiple hypergeometric functions of the Kampé de Fériet are derived. In addition, mixed relations (80), (82) and (84) involving the Riemann zeta functions are also established.

We study the partition norm—the product of the parts of a partition—with emphasis on partitions whose parts all exceed 1. We obtain two bivariate recurrences for the sum of norms over partitions into distinct parts, including a refinement that separates the contributions of the part 1 and of the prime 2. To count partitions of n with parts greater than 1 having norm i, we introduce the restricted norm-counting function r_{>1}(i,n), give its two-parameter generating function, and derive recurrences and other relations. Finally, we formulate analogues of Goldbach’s and twin-prime conjectures in the language of partition norms.

In this paper, we introduce q-generalized harmonic numbers with two parameters ω and ξ, H_{μ,λ,q}(ω,ξ) for integers μ, λ such that λ ≥ μ. With the help of these numbers, we define a new family of numbers which is called q-generalized hyperharmonic numbers with two parameters ω and ξ of order r, H_{μ,λ,q}^{r}(ω,ξ) for integer r. Then, we consider special matrices whose entries are given by these numbers and give some matrix multiplications. Additionally, we derive some combinatorial identities for H_{μ,λ,q}(ω,ξ) and H_{μ,λ,q}^{r}(ω,ξ) by matrix methods.

Let $\lambda_{{\rm sym}^{r}f}(n)$ be the n-th coefficient in the Dirichlet series representing the symmetric power L-function attached to a primitive form f of weight k and level N. In this paper, we give asymptotic formulas for the arithmetic mean of $\lambda_{{\rm sym}^{r}f}(n)$ on integers under digital constraints related to the sum of digits function. Our results throw the light on the behavior of the classical function $\lambda_{{\rm sym}^{r}f}(n)$ on integers in arithmetic progression related to the sum of digits function.

In this paper, we define two new sequences using the Fibonacci p-numbers, the generalized Mersenne numbers, and m-balancing numbers. These sequences are obtained from the corresponding characteristic polynomials. The determinants and combinatorial and exponential representations of these new sequences are given. As an application, we provide two algorithms using these new sequences with self-invertible matrices and the Affine–Hill cipher.

In the previous author's research, the set of $\underline{\rm Set}(n)$'s for natural numbers $n$ was constructed. For this set it was proved that it is a commutative semi-group. The condition for which it is a monoid was given. The present leg of research continues by demonstrating that for any $n,$ $\underline{\rm Set}(n)$ is a lattice, and by the introduction of four new operations over the elements of $\underline{\rm Set}(n)$.

The power Fibonacci sequence in $\mathbb{Z}_m[\sqrt{\delta}]$ is defined as a Fibonacci sequence \linebreak $F_n = F_{n-1} + F_{n-2}$ where $F_0 = 1$ and $F_1 = a $, such that $a\in \mathbb{Z}_m[\sqrt{\delta}]$ and $F_n\equiv a^n \:(\mathrm{mod}\: m)$, for all $n\in \mathbb{N}\cup \{0\}$. In this paper, we investigated the existence of power Fibonacci sequences in $\mathbb{Z}_m[\sqrt{\delta}]$, and the number of power Fibonacci sequences in $\mathbb{Z}_m[\sqrt{\delta}]$ for a given $m$, where $\delta$ is a square-free integer. Furthermore, we determined explicitly all power Fibonacci sequences in $\mathbb{Z}_{p^k}[\sqrt{\delta}]$, where $p$ is a prime number.

We revisit a classical theorem of Chebyshev about distribution of primes on intervals $(n, 2n)$ $n\in\mathbb N$, and prove a generalization of it. Extending Erdős' arithmetical-combinatorial argument, we show that for all $k\in\mathbb N$, there is $n_k\in\mathbb N$ such that the intervals $(kn, (k+1)n)$ contain a prime for all $n\geq n_k$. A quantitative lower bound is derived for the number of primes on such intervals. We also give numerical upper bounds for $n_k$ for $k \leq 20$, and we draw comparisons with existing results in the literature.