Related papers: How many digits are needed?

Define a natural number $n$ as a \textit{square-full} integer if for every prime $p$ such that $p|n$, we have $p^2|n$. In this paper, we establish an upper bound on the variance of square-full integers in short intervals of an expected…

We deal with a sequence of integer-valued random variables $\{Z_N\}_{N=1}^{\infty}$ which is related to restricted partitions of positive integers. We observe that $Z_N=X_1+ \ldots + X_N$ for independent and bounded random variables…

We provide conditions on dependent and on non-stationary random variables $X_n$ ensuring that the mantissa of the sequence of products $\left(\prod_{1}^{n}X_k\right)$ is almost surely distributed following the Benford's law or converges in…

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the…

We study a new family of random variables, that each arise as the distribution of the maximum or minimum of a random number $N$ of i.i.d.~random variables $X_1,X_2,\ldots,X_N$, each distributed as a variable $X$ with support on $[0,1]$. The…

Sums of of 1-dependent integer-valued random variables are approximated by compound Poisson, negative binomial and Binomial distributions and signed compound Poisson measures. Estimates are obtained for total variation and local metrics.…

In this paper, we consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law cannot be expected. The setting is one in which the simplest approximation to the…

By a classical result of Gauss and Kuzmin, the continued fraction expansion of a ``random'' real number contains each digit $a\in\mathbb{N}$ with asymptotic frequency $\log_2(1+1/(a(a+2)))$. We generalize this result in two directions:…

Nonextensive statistics, characterized by a nonextensive parameter $q$, is a promising and practically useful generalization of the Boltzmann statistics to describe power-law behaviors from physical and social observations. We here explore…

A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore,…

The \textit{ternary digits of $2^n$} are a finite sequence of 0s, 1s, and 2s. It is a natural question to ask whether the frequency of any string of 0s, 1s, and 2s in this sequence approaches the same limit for all strings of the same…

Let $G=(G_j)_{j\ge 0}$ be a strictly increasing linear recurrent sequence of integers with $G_0=1$ having characteristic polynomial $X^{d}-a_1X^{d-1}-\cdots-a_{d-1}X-a_d$. It is well known that each positive integer $\nu$ can be uniquely…

A subset A of {0,1,...,n} is said to be a 2-additive basis for {1,2,...,n} if each j in {1,2,...,n} can be written as j=x+y, x,y in A, x<=y. If we pick each integer in {0,1,...,n} independently with probability p=p_n tending to 0, thus…

The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$ is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a…

In this paper, we study the Lebesgue structure of the distribution of a random variable given in terms of a continued fraction with a two-symbol alphabet $\{\frac{1}{2}, 1\}$, also known as $A_2$-fractions. We establish necessary and…

We consider integers whose squares have just three decimal digits. Examples are e.g. given by $2108436491907081488939581538^2 = 4445504440405440505004450045555054500055550554550445444$ and $10100000000010401000000000101^2 =…

Let $s\_2(x)$ denote the number of digits "$1$" in a binary expansion of any $x \in \mathbb{N}$. We study the mean distribution $\mu\_a$ of the quantity $s\_2(x+a)-s\_2(x)$ for a fixed positive integer $a$.It is shown that solutions of the…

We study "dead ends" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite…

This paper deals with empirical processes of the type \[C_n(B)=\sqrt{n}\{\mu_n(B)-P(X_{n+1}\in B\mid X_1,...,X_n)\},\] where $(X_n)$ is a sequence of random variables and $\mu_n=(1/n)\sum_{i=1}^n\delta_{X_i}$ the empirical measure.…

A $\widetilde{Q}-$representation of real numbers is introduced as a generalization of the $p-$adic and $Q-$representations. It is shown that the $\widetilde{Q}-$representation may be used as a convenient tool for the construction and study…