Related papers: The Smallest Solution of \phi(30n+1)<\phi(30n) is …

Define $|n|$ to be the complexity of $n$, the smallest number of 1's needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $|n|\ge 3\log_3 n$ for all $n$. Define the defect of $n$,…

We consider the \emph{smallest superpolyomino problem}: given a set of colored polyominoes, find the smallest polyomino containing each input polyomino as a subshape. This problem is shown to be NP-hard, even when restricted to a set of…

Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer…

For any positive integer $n$, let $\sigma (n)$ be the sum of all positive divisors of $n.$ In this paper, it is proved that the set of positive integers $ n $ for which $ \sigma(30n+1)\geq \sigma(30n) $ has a density less than $ 0.0371813,…

Based on the first 25 known values of Pi(10^n), the number of primes less than 10^n, with n integer between 1 and 25, we propose a conjectured value range of Pi(10^26) calculated by using polynomial interpolations with two corrective…

A recent breakthrough in computer-assisted mathematics showed that every set of $30$ points in the plane in general position (i.e., without three on a common line) contains an empty convex hexagon, thus closing a line of research dating…

In this paper, we give a new upper bound for the number $N_{\mathcal{R}}$ which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function $\pi(x)$ holds for every $x…

Let $I(n) = \frac{\psi(\phi(n))}{\phi(\psi(n))}$ and $K(n) = \frac{\psi(\phi(n))}{\phi(\phi(n))}$, where $\phi(n)$ is Euler's function and $\psi(n)$ is Dedekind's arithmetic function. We obtain the maximal order of $I(n)$, as well as the…

We give a closed-form expression for $\varphi(1+\varphi(2+\varphi(3+...+\varphi(n)$, where $\varphi$ is Euler's totient function. More generally, for an integer sequence $A=\{a_j\}$ we study the value of…

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a…

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the…

We study a sequence of differences related to the problem of finding the smallest factorial $n!$ greater than or equal to $a^n$, where $a > 1$, using the gamma function.

In this work, first we prove that for any compact set $K\subset\mathbb{R}^{n}$ and any continuous function $\phi$ defined on $\partial K$, there exists a bounded weak solution in $C(\bar{\mathbb{R}^{n}\backslash K}) \cap…

Let $\phi(n)$denote Euler's phi function. We study the distribution of the numbers $gcd(n,\phi(n))$ and their divisors. Our results generalize previous results of Erd\H{o}s and Pollack.

For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by $\sum_{k=0}^{[n/2]}\binom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$, where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is the Euler number, $E_n^{(a)}$…

In this paper we present 43 new inequalities related to integer part and fractional part.

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(\pi(R_n))_{n \in \mathbb{N}}$, which…

Each family of Gauss hypergeometric functions $$ f_n={}_2F_1(a+\epsilon_1n, b+\epsilon_2n ;c+\epsilon_3n; z), $$ for fixed $\epsilon_j=0,\pm1$ (not all $\epsilon_j$ equal to zero) satisfies a second order linear difference equation of the…

Let $M(x)$ denote the largest cardinality of a subset of $\{n \in \mathbf{N}: n \leq x\}$ on which the Euler totient function $\varphi(n)$ is non-decreasing. We show that $M(x) = (1+O(\frac{(\log\log x)^5}{\log x})) \pi(x)$ for all $x \geq…

When $k>1$ and $n$ is the product of the smallest $k$ primes, the $(k+1)$-st smallest prime is the least divisor exceeding $1$ of $n^{n^n}-1$. This variant of Euclid's prime generator is discussed with some of its cousins.