Related papers: On 2-Near Perfect Numbers

It is an open problem whether $ \binom{2n}{n} $ is divisible by 4 or 9 for all $n>256$. In connection with this, we prove that for a fixed uneven $m$ the asymptotic density of $k$'s such that $ m \nmid \binom{2^{k+1}}{2^{k}} $ is 0. To do…

If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to…

Let $A$ be a set of $n$ positive integers. We say that a subset $B$ of $A$ is a divisor of $A$, if the sum of the elements in $B$ divides the sum of the elements in $A$. We are interested in the following extremal problem. For each $n$,…

A polynomial is a direct sum if it can be written as a sum of two non-zero polynomials in some distinct sets of variables, up to a linear change of variables. We analyze criteria for a homogeneous polynomial to be decomposable as a direct…

We shall show that there is no odd perfect number of the form $2^n+1$ or $n^n+1$.

In this paper, we study the distribution of the sequence of integers $2^{\omega(n)}$ under the assumption of the strong Riemann hypothesis, where $\omega(n)$ denotes the number of distinct prime divisors of $n$. We provide an asymptotic…

We will show the two following results: If there existe an odd perfect number $n$ of prime decomposition $n=p_1^{\alpha_1} \ldots p_k^{\alpha_k}q^\beta$, where the $\alpha_i$ are even, the $\beta$ are odd and $q \equiv 5 \mod 8$. Then there…

This paper is concerned with the problem of finding two sets of integers, $\{a_1, a_2, \ldots$, $a_m\}$ and $\{b_1, b_2, \ldots, b_n\}$, such that all the $mn$ sums $a_i+b_j, i=1, \ldots, m, j=1, \ldots, n$, are perfect squares. A method is…

In this paper we discuss near-perfect numbers of various forms. In particular, we study the existence of near-perfect numbers in the Fibonacci and Lucas sequences, near-perfect values taken by integer polynomials and repdigit near-perfect…

Let $p^k m^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $\sigma(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application…

A subset of $\{1,2,\ldots,2n\}$ is said to be primitive if it does not contain any pair of elements $(u,v)$ such that $u$ is a divisor of $v$. Let $D(n)$ denote the number of primitive subsets of $\{1,2,\ldots,2n\}$ with $n$ elements.…

Let \sigma(n) be the sum of divisors of a positive integer n. Robin's theorem states that the Riemann hypothesis is equivalent to the inequality \sigma(n)<e^\gamma n\log\log n for all n>5040 (\gamma is Euler's constant). It is a natural…

A positive integer $n$ is called an abundant number if $\sigma (n)\ge 2n$, where $\sigma (n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H…

Much recent progress has been made concerning the probable existence of Odd Perfect Numbers, forming part of what has come to be known as Sylvester's Web Of Conditions. This paper proves some results concerning certain properties of the…

We study the problem of finding positive integers $n$ such that all the decimal digits of $2^n$ are even, i.e., belong to $\{0, 2, 4, 6, 8\}$. Computational checks up to $n = 10^{15}$ reveal the known cases $n = 1, 2, 3, 6, 11$ and no…

In this paper, we prove that the 2-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of 2- factors that contain the the perfect matching as a subgraph. Consequently, we show that the…

New Mersenne conjectures. The problems of simplicity, common prime divisors and free from squares of numbers $L(n) = 2^{2n}\pm2^n\pm1$ are investigated. Wonderful formulas $gcd $ for numbers $L (n) $ and numbers repunit are proved.

An odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. Similarly, an even perfect number $M$ is said to be given in Euclidean form if $M…

If $s$ is a positive integer and $A$ is a set of positive integers, we say that $B$ is an $s$-divisor of $A$ if $\sum_{b\in B} b\mid s\sum_{a\in A} a$. We study the maximal number of $k$-subsets of an $n$-element set that can be…

In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\ldots$ define $$d_n(x)=\sum_{k=0}^n\binom nk\binom xk2^k$$ and $$s_n(x)=\sum_{k=0}^n\binom nk\binom xk\binom{x+k}k=\sum_{k=0}^n\binom…