Add to ORKGLet $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second smallest, third smallest and fourth smallest prime factors. We also obtain tighter bounds for odd perfect numbers. We also discuss the behavior of $\sigma(n!+1)$, $\sigma(2^n+1)$, and related sequences.Comment: 19 pages. Accepted to Integer
We show that the existence of arithmetic progressions with few primes, with a quantitative bound on ...
http://www.math.missouri.edu/~bbanks/papers/index.htmlLet '(•) and _(•) denote the Euler function an...
AbstractIn this paper we prove that if (r,12)⩽3, then the set of positive odd integers k such that k...
Let $\sigma(n)$ be the sum of the positive divisors of $n$. A number $n$ is non-deficient if $\sigma...
Let $n$ be a primitive non-deficient number where $n=p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1...
We say n ∈ ℕ is perfect if σ (n) = 2n, where σ(n) denotes the sum of the positive divisors of n. No ...
In this work we construct a lower bound for an odd perfect number in terms of the number of its dist...
AbstractIt is not known whether or not there exists an odd perfect number. We describe an algorithmi...
Inspired by a classical result of R\'enyi, we prove that every even integer $N\geq 4$ can be written...
In this work we construct a lower bound for an odd perfect number in terms of the number of its dist...
Let $p^k m^2$ be an odd perfect number with special prime $p$. Extending previous work of the author...
This short note provides a sharper upper bound of a well known inequality for the sum of divisors fu...
For a positive integer $n$, let $\sigma(n)$ denote the sum of the positive divisors of $n$, and le...
Abstract. Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is sai...
Abstract. Let σ(n) denote the sum of the positive divisors of n. We say that n is perfect if σ(n) =...
We show that the existence of arithmetic progressions with few primes, with a quantitative bound on ...
http://www.math.missouri.edu/~bbanks/papers/index.htmlLet '(•) and _(•) denote the Euler function an...
AbstractIn this paper we prove that if (r,12)⩽3, then the set of positive odd integers k such that k...
Let $\sigma(n)$ be the sum of the positive divisors of $n$. A number $n$ is non-deficient if $\sigma...
Let $n$ be a primitive non-deficient number where $n=p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1...
We say n ∈ ℕ is perfect if σ (n) = 2n, where σ(n) denotes the sum of the positive divisors of n. No ...
In this work we construct a lower bound for an odd perfect number in terms of the number of its dist...
AbstractIt is not known whether or not there exists an odd perfect number. We describe an algorithmi...
Inspired by a classical result of R\'enyi, we prove that every even integer $N\geq 4$ can be written...
In this work we construct a lower bound for an odd perfect number in terms of the number of its dist...
Let $p^k m^2$ be an odd perfect number with special prime $p$. Extending previous work of the author...
This short note provides a sharper upper bound of a well known inequality for the sum of divisors fu...
For a positive integer $n$, let $\sigma(n)$ denote the sum of the positive divisors of $n$, and le...
Abstract. Let σ(n) denote the sum of positive divisors of the natural number n. Such a number is sai...
Abstract. Let σ(n) denote the sum of the positive divisors of n. We say that n is perfect if σ(n) =...
We show that the existence of arithmetic progressions with few primes, with a quantitative bound on ...
http://www.math.missouri.edu/~bbanks/papers/index.htmlLet '(•) and _(•) denote the Euler function an...
AbstractIn this paper we prove that if (r,12)⩽3, then the set of positive odd integers k such that k...