Related papers: Odd Multiperfect Numbers

An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 does not divide N, then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect…

Let $\sigma(x)$ be the sum of the divisors of $x$. If $N$ is odd and $\sigma(N) = 2N$, then the 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…

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…

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 +…

A natural number $n$ is said to be $k$- multiperfect number if $\sigma (n) = k\cdot n$ for some integer $k>2$. In this paper, I will provide a lower bound for $\tau(n)$ of any $k$- multiperfect numbers. The lower bound for $\tau(n)$ will…

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…

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…

We generalize the definition of spoof perfect numbers to multiperfect numbers and study their characteristics. As a result, we find several new odd spoof multiperfect numbers, akin to Descartes' number. An example is $8999757$, which would…

Let $\sigma(n)$ be the sum of the divisors of $n$. Kalita and Saikia defined a number $n$ to be near superperfect if $2n+d=\sigma(\sigma(n))$ for some positive divisor $d$ of $n$. We extend some of their results about near superperfect…

We show that for any natural number $n$ satisfying $n\equiv 4 \mod 8$ and $n\not\equiv 0 \mod 5$, and for any odd integer $t\geq \frac{n+6}{2}$ there are infinitely many Salem numbers ${\alpha}$ of degree $2t$ such that ${\alpha}^n-1$ is a…

Let $n$ and $k$ be positive integers and $\sigma(n)$ the sum of all positive divisors of $n$. We call $n$ an exactly $k$-deficient-perfect number with deficient divisors $d_1, d_2, \ldots, d_k$ if $d_1, d_2, \ldots, d_k$ are distinct proper…

If $N = {p^k}{m^2}$ is an odd perfect number with special prime factor $p$, then it is proved that ${p^k} < (2/3){m^2}$. Numerical results on the abundancy indices $\frac{\sigma(p^k)}{p^k}$ and $\frac{\sigma(m^2)}{m^2}$, and the ratios…

We study the set $\mathcal{S}$ of odd positive integers $n$ with the property ${2n}/{\sigma(n)} - 1 = 1/x$, for positive integer $x$, i.e., the set that relates to odd perfect and odd "spoof perfect" numbers. As a consequence, we find that…

Euler showed that if an odd perfect number $N$ exists, it must consist of two parts $N=q^k n^2$, with $q$ prime, $q \equiv k \equiv 1 \pmod{4}$, and gcd$(q,n)=1$. Dris conjectured that $q^k < n$. We first show that $q<n$ for all odd perfect…

Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $\sigma(n) = 2 n$. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form…

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 $I(q^k) < I(n)$,…

Some new results concerning the equation $\sigma(N)=aM, \sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.

Let $k>2$ be a prime such that $2^k-1$ is a Mersenne prime. Let $n = 2^{\alpha-1}p$, where $\alpha>1$ and $p<3\cdot 2^{\alpha-1}-1$ is an odd prime. Continuing the work of Cai et al. and Jiang, we prove that $n\ |\ \sigma_k(n)$ if and only…

We shall prove that if $N=p^\alpha q_1^{2\beta_1} q_2^{2\beta_2} \cdots q_{r-1}^{2\beta_{r-1}}$ is an odd perfect number such that $p, q_1, \ldots, q_{r-1}$ are distinct primes, $p\equiv\alpha\equiv 1\mod{4}$ and $t$ divides $2\beta_i+1$…

Motivated by the works of Erd\"os, Pomerance, Wolke and Harman on the sum-of-divisor function $\sigma(n)$, we study the distribution of a special class of natural numbers closely related to (multiply) perfect numbers which we term…