Related papers: Smarandache Function Applied to Perfect Numbers

In this thesis we examined mathematical properties of Fibonacci numbers and applications of this numbers in the nature,geometry and economy.We obtained Golden section and proved some mathematical identities using Golden section. Infinity of…

A positive integer $n$ is called perfect if $ \sigma(n)=2n$, where $\sigma(n)$ denote the sum of divisors of $n$. In this paper we study the ratio $\frac{\sigma(n)}{n}$. We define the function Abundancy Index $I:\mathbb{N} \to \mathbb{Q}$…

A perfect number is a number whose divisors add up to twice the number itself. The existence of odd perfect numbers is a millennia-old unsolved problem. This note proposes a proof of the nonexistence of odd perfect numbers. More generally,…

We adapt (over $\mathbb{F}_2$) the general notions of multiplicative function, Dirichlet convolution and Inverse. We get some interesting results, namely necessary conditions for an odd binary polynomial to be perfect. Note that we are…

We investigate a result on convergence of double sequences of numbers and how it extends to measurable functions.

We present some formulas for the computation of the zeros of the integral-degree associated Legendre functions with respect to the order.

In this paper, we study integral functionals defined on spaces of functions with values on general (non-separable) Banach spaces. We introduce a new class of integrands and multifunctions for which we obtain measurable selection results.…

We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.

We give necessary conditions for perfection of some families of odd numbers with special multiplicative forms. Extending earlier work of Steuerwald, Kanold, McDaniel et al.

In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.

Through an inversion approach, we suggest a possible estimation for the absolute value of Mertens function $\vert M(x) \vert$ that $ \left\vert M(x) \right\vert \sim \left[\frac{1}{\pi \sqrt{\varepsilon}(x+\varepsilon)}\right]\sqrt{x}$…

We study the equation $F_n + F_m = y^p$, where $F_n$ and $F_m$ are respectively the $n$-th and $m$-th Fibonacci numbers and $p \ge 2$. We find all solutions under the assumption $n \equiv m \pmod{2}$.

In the integer case, the Smarandache function of a positive integer $n$ is defined to be the smallest positive integer $k$ such that $n$ divides the factorial $k!$. In this paper, we first define a natural order for polynomials in…

We show that $n$ is almost perfect if and only if $I(n) - 1 < D(n) \leq I(n)$, where $I(n)$ is the abundancy index of $n$ and $D(n)$ is the deficiency of $n$. This criterion is then extended to the case of integers $m$ satisfying $D(m)>1$.

A perfect number is a positive integer n such that n equals the sum of all positive integer divisors of n that are less than n. That is, although n is a divisor of n, n is excluded from this sum. Thus 6 = 1 + 2 + 3 is perfect, but 12 < 1 +…

We reveal a relationship between the prime counting function and an operation performed on a unique subsequence of the primes.

We propose the construction of entire functions with a given random collection of zeros. There are considered two particular cases. In the first one we are dealing with simple zeros. And the second corresponds to random zeros with random…

We consider several possible approaches to evaluating an integral involving the digamma function and a related logarithmic series.

We consider the functional inverse of the Gamma function in the complex plane, where it is multi-valued, and define a set of suitable branches by proposing a natural extension from the real case.

We introduce a total order and the absolute value function for dual numbers. The absolute value function of dual numbers are with dual number values, and have properties similar to the properties of the absolute value function of real…