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Register a new userThe reciprocal sum of divisors of Mersenne numbers
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We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine $max_{nle x} sum_{p mid 2^n-1} 1/p$ to within $o(1)$ and $max_{nle x} sum_{dmid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $xtoinfty$. This refines, conditionally, earlier estimates of ErdH{o}s and ErdH{o}s-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $sum 1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing $2^n-1$ for some $nle x$. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both $sum_{pmid 2^n-1} 1/p$ and $sum_{dmid 2^n-1}1/d$ admit continuous distribution functions in the sense of probabilistic number theory.
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We consider the distribution in residue classes modulo primes $p$ of Eulers totient function $phi(n)$ and the sum-of-proper-divisors function $s(n):=sigma(n)-n$. We prove that the values $phi(n)$, for $nle x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 le p le (log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $ple (log{x})^A$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with $x$.
In this research paper, relationship between every Mersenne prime and certain Natural numbers is explored. We begin by proving that every Mersenne prime is of the form {4n + 3,for some integer n} and generalize the result to all powers of 2. We also tabulate and show their relationship with other whole numbers up to 10. A number of minor results are also proved. Based on these results, approaches to determine the cardinality of Mersenne primes are discussed.
In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The so-called binomial sums are also considered. The problem of constructing polynomials that allow to calculate the values of the corresponding sums in certain cases is solved.
By $(mathbb{Z}^+)^{infty}$ we denote the set of all the infinite sequences $mathcal{S}={s_i}_{i=1}^{infty}$ of positive integers (note that all the $s_i$ are not necessarily distinct and not necessarily monotonic). Let $f(x)$ be a polynomial of nonnegative integer coefficients. Let $mathcal{S}_n:={s_1, ..., s_n}$ and $H_f(mathcal{S}_n):=sum_{k=1}^{n}frac{1}{f(k)^{s_{k}}}$. When $f(x)$ is linear, Feng, Hong, Jiang and Yin proved in [A generalization of a theorem of Nagell, Acta Math. Hungari, in press] that for any infinite sequence $mathcal{S}$ of positive integers, $H_f(mathcal{S}_n)$ is never an integer if $nge 2$. Now let deg$f(x)ge 2$. Clearly, $0<H_f(mathcal{S}_n)<zeta(2)<2$. But it is not clear whether the reciprocal power sum $H_f(mathcal{S}_n)$ can take 1 as its value. In this paper, with the help of a result of ErdH{o}s, we use the analytic and $p$-adic method to show that for any infinite sequence $mathcal{S}$ of positive integers and any positive integer $nge 2$, $H_f(mathcal{S}_n)$ is never equal to 1. Furthermore, we use a result of Kakeya to show that if $frac{1}{f(k)}lesum_{i=1}^inftyfrac{1}{f(k+i)}$ holds for all positive integers $k$, then the union set $bigcuplimits_{mathcal{S}in (mathbb{Z}^+)^{infty}} { H_f(mathcal{S}_n) | nin mathbb{Z}^+ }$ is dense in the interval $(0,alpha_f)$ with $alpha_f:=sum_{k=1}^{infty}frac{1}{f(k)}$. It is well known that $alpha_f= frac{1}{2}big(pi frac{e^{2pi}+1}{e^{2pi}-1}-1big)approx 1.076674$ when $f(x)=x^2+1$. Our dense result infers that when $f(x)=x^2+1$, for any sufficiently small $varepsilon >0$, there are positive integers $n_1$ and $n_2$ and infinite sequences $mathcal{S}^{(1)}$ and $mathcal{S}^{(2)}$ of positive integers such that $1-varepsilon<H_f(mathcal{S}^{(1)}_{n_1})<1$ and $1<H_f(mathcal{S}^{(2)}_{n_2})<1+varepsilon$.
Let $s(n):= sum_{dmid n,~d<n} d$ denote the sum of the proper divisors of $n$. It is natural to conjecture that for each integer $kge 2$, the equivalence [ text{$n$ is $k$th powerfree} Longleftrightarrow text{$s(n)$ is $k$th powerfree} ] holds almost always (meaning, on a set of asymptotic density $1$). We prove this for $kge 4$.
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