Related papers: A note on odd perfect numbers

Acquaah and Konyagin showed that if $N$ is an odd perfect number where $N= p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1 < p_2 \cdots < p_k$ then one must have $p_k < 3^{1/3}N^{1/3}$. Using methods similar to theirs, we show that…

Let $N$ be an odd perfect number. Let $\omega(N)$ be the number of distinct prime factors of $N$ and let $\Omega(N)$ be the total number (counting multiplicity) of prime factors of $N$. We prove that $\frac{99}{37}\omega(N) - \frac{187}{37}…

In this note, we obtain an exact formula for the $q$-adic valuation of $\sigma_k(n)$ where $q$ is an odd prime, allowing us to derive an explicit upper bound which is asymptotically better than the previous bound obtained by Zhao when $n$…

We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{\alpha_1}\cdots p_s^{\alpha_s} q_1^{\beta_1}\cdots q_t^{\beta_t}$ with $\beta_1, \ldots, \beta_t$ bounded by a given constant.…

If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $n < q$ is sufficient for Sorli's conjecture that $k = \nu_{q}(N) = 1$ to hold. We also prove that $q^k < 2/3{n^2}$, and that $I(q^k) < I(n)$,…

Let $\sigma(n)$ be the sum of the positive divisors of $n$. A positive integer $n$ is said to be $2$-near perfect when $\sigma(n)=2n+d_1+d_2$, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We show that there are no odd…

The Euler's form of odd perfect numbers, if any, is $n=\pi^{\alpha}N^2$, where $\pi$ is prime, $(\pi,N)=1$ and $\pi\equiv \alpha \equiv 1 \pmod{4}$. Dris conjecture states that $N>\pi^{\alpha}$. We find that $N^2>\frac{1}{2}\pi^{\gamma}$,…

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…

Let G be a finite group and let k be a positive integer. We examine the relationship between structural properties of G and the number of elements of G that are not kth powers in G. In particular, we examine a bound on |G| given by Lucido…

We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of at most six primes. We follow the circle…

Solitary numbers are shrouded with mystery. A folklore conjecture assert that 10 is a solitary number i.e. it has no friends. In this article, we establish that if $N$ is a friend of $10$ then it must be odd square with at least seven…

We shall given a new effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent, improving our old result in T. Yamada, Colloq. Math. 103 (2005), 303--307.

Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as…

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 call $n$ a spoof odd perfect number if $n$ is odd and $n=km$ for two integers $k,m>1$ such that $\sigma(k)(m+1)=2n$, where $\sigma$ is the sum-of-divisors function. In this paper, we show how results analogous to those of odd perfect…

Let $q^k n^2$ be an odd perfect number with special prime $q$. Define the GCDs $$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$ $$H = \gcd\bigg(n^2,\sigma(n^2)\bigg)$$ and $$I = \gcd\bigg(n,\sigma(n^2)\bigg).$$ We prove that $G \times H =…

It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer $N=(4k+1)^{4m+1}\prod_{i=1}^\ell ~ q_i^{2\alpha_i}$ to establish that there do…

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

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 $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then Sorli's conjecture predicts that $k=\nu_{q}(N)=1$. In this article, we give some further results related to this conjecture and those contained in the papers…