Related papers: Sequence elimination function and the formulas of …
Necessary and sufficient conditions under which two real functions defined on the real interval can be separated by a polynomial are given. An immediate consequence of the main result is the existence of the polynomial separation of convex…
Let $A$ be a nonempty finite set of $k$ integers. Given a subset $B$ of $A$, the sum of all elements of $B$, denoted by $s(B)$, is called the subset sum of $B$. For a nonnegative integer $\alpha$ ($\leq k$), let \[\Sigma_{\alpha}…
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
In this note we describe a method for finding prime numbers as fixed points of particularly simple sequences. Some basic calculations show that success rates for identifying primes this way are over 99.9%. In particular, it seems that the…
Let $\mathbf{S}$ be the set of all finite or infinite increasing sequences of positive integers. For a sequence $S=\{s(n)\}, n\geq1,$ from $\mathbf{S},$ let us call a positive number $N$ an exponentially $S$-number $(N\in E(S)),$ if all…
We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a…
Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. The aim of this article is to give a result about the sum of euler's totient function from k equal 1 to n whene p divides n and p…
We consider the avoidance of patterns in inversion sequences that relate sorting via sorting machines including data structures such as pop stacks and stacks. Such machines have been studied under a variety of additional constraints and…
Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…
In this study, several interesting iterative sequences were investigated. First, we define the iterative sequences. We fix function f(n). An iterative sequence starts with a natural number n, and calculates the sequence f(n),f(f(n)),…
We define a function by refining Stern's diatomic sequence. We name it the {\it assembly function}. It is strictly increasing continuous. The first and the second main theorems are on an action to the function. The third theorem is on…
Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.
We define a proper differential sequence of ordinary differential equations and introduce a method to derive an alternative sequence of integrals for such a sequence. We describe some general properties which are illustrated by several…
We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases…
The $\Sopfr(n)$ function is defined as the sum of prime factors of $n$ each of which is taken with its multiplicity. This function is studied numerically. The analogy between $\Sopfr(n)$ and the primes distribution function is drawn and…
We introduce an algorithmic approach based on generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate…
New exceptional (i.e. non-repeating) prime number multiplets are given and formulated in terms of arithmetic progressions, along with laws governing them. Accompanying repeating prime number multiplets are pointed out. Prime number…
Given a barrier $0 \leq b_0 \leq b_1 \leq ...$, let $f(n)$ be the number of nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq ... \leq a_n$ for which $a_j \leq b_j$ for all $0 \leq j \leq n$. Known formul\ae for $f(n)$ include an $n…
Working in a semi-constructive logical system that supports the extraction of concurrent programs, we extract a program inverting non-singular real valued matrices from a constructive proof based on Gaussian elimination. Concurrency is used…
Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.