Related papers: Smarandache Function Applied to Perfect Numbers
In this short article we present some properties regarding the order and the type of an entire function.
We determine all perfect powers that can be written as the sum of at most 10 consecutive squares.
In this paper we address the problem of feature selection when the data is functional, we study several statistical procedures including classification, regression and principal components. One advantage of the blinding procedure is that it…
In this paper, we introduce the concept of $F$-perfect number, which is a positive integer $n$ such that $\sum_{d|n,d<n}d^2=3n$. We prove that all the $F$-perfect numbers are of the form $n=F_{2k-1}F_{2k+1}$, where both $F_{2k-1}$ and…
In this paper, we introduce a new generalization of the perfect numbers, called $\mathcal{S}$-perfect numbers. Briefly stated, an $\mathcal{S}$-perfect number is an integer equal to a weighted sum of its proper divisors, where the weights…
In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation {\eta}({\pi}(x)) = {\pi}({\eta}(x)), where {\eta} is the Smarandache function and {\pi}…
We prove a law of large numbers in terms of complete convergence of independent random variables taking values in increments of monotone functions, with convergence uniform both in the initial and the final time. The result holds also for…
This article discusses some important applications of the quadratic function with the aim of highlighting the importance of cuadr\'aticas.- forms are also intended to show how a simple function covers virtually all areas of knowledge are…
In this paper we give a novel solution to a classical completion problem for square matrices. This problem was studied by many authors through time, and it is completely solved in [2, 3]. In this paper we relate this classical problem to a…
In this paper we determine the perfect powers that are sums of three fifth powers in an arithmetic progression. More precisely, we completely solve the Diophantine equation $$ (x-d)^5 + x^5 + (x + d)^5 = z^n,~n\geq 2, $$ where $d,x,z \in…
The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the…
We introduce a new factorial function which agrees with the usual Euler gamma function at both the positive integers and at all half-integers, but which is also entire. We describe the basic features of this function.
In this research, an optimal algorithm for the Collatz conjecture is presented. Properties such as the convergence of the algorithm and an equation that relates the algorithm to the classical Collatz conjecture are obtained. It is validated…
We investigate uniqueness problems for an entire function that shares two small functions of finite order with their difference operators. In particular, we give a generalization of a result in $[2]$.
We review the results having the property of maximal transcendentality.
We derive new reduction formulas for the incomplete beta function and the Lerch transcendent in terms of elementary functions. As an application, we calculate some new integrals. Also, we use these reduction formulas to test the performance…
In this article we propose a general method of obtaining infinite sums of products with functions that count patterns in numbers.
We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.
We review some results of calculations, having the property of maximal transcendentality.
The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…