The non-existence of odd perfect numbers of a certain form

  title={The non-existence of odd perfect numbers of a certain form},
  author={Wayne L. McDaniel},
  journal={Archiv der Mathematik},
Odd multiperfect numbers of abundancy 4
Explicit sieve estimates and nonexistence of odd multiperfect numbers of a certain form
. We prove explicit asymptotic formulae for some functions used in sieve methods and show that there exists no odd multiperfect number of abundancy four whose squared part is cubefree.
The notion oftotient, super totient and hyper totient numbers are introduced and their relations are discussed and applications in graph labeling have been demonstrated over a family of well known graph.
A new upper bound for odd perfect numbers of a special form
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),
Quasiperfect Numbers With the Same Exponent
A general lower bound for the smallerst prime factor of quasiperfect numbers is found and some upper bounds concerning quas Piperfect numbers of the form $N=m^2$ with $m$ squarefree are found.
An exponential diophantine equation related to odd perfect numbers
We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give
On the divisibility of odd perfect numbers, quasiperfect numbers and amicable numbers by a high power of a prime
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,
More on the Nonexistence of Odd Perfect Numbers of a Certain Form
Euler showed that if an odd perfect number exists, it must be of the form $N = p^\alpha q_{1}^{2\beta_{1}}$ $\ldots$ $q_{k}^{2\beta_{k}}$, where $p, q_{1}, \ldots, q_k$ are distinct odd primes,
Sieve methods for odd perfect numbers
It is proved that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 10^8 < p < 10^1000.
Odd multiperfect numbers
A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for integer $k\ge2$ if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the positive divisors of $n$. In this paper, we


Introduction to Number Theory
A specific feature of this text on number theory is the rather extensive treatment of Diophantine equations of second or higher degree. A large number of non-routine problems are given. The book is