Related papers: Random index of codivisibility
The abundancy index of a positive integer is the ratio between the sum of its divisors and itself. We generalize previous results on abundancy indices by defining a two-variable abundancy index function as…
We consider procedures of sampling parts from a random integer partition. We determine asymptotically the probabilty distribution of the randomly-selected part whenever the positive integer that is partitioned becomes large.
For a pair of random Gaussian integers chosen uniformly and independently from the set of Gaussian integers of norm $x$ or less as $x$ goes to infinity, we find asymptotics for the average norm of their greatest common divisor, with…
In a sample variance decomposition, with components functions of the sample's spacings, the largest component $\tilde{I}_n$ is used in cluster detection. It is shown for normal samples that the asymptotic distribution of $\tilde{I}_n$ is…
A pattern of a sequence is a sequence of integer indices with each index describing the order of first occurrence of the respective symbol in the original sequence. In a recent paper, tight general bounds on the block entropy of patterns of…
Abundancy index refers to the ratio of the sum of the divisors of a number to the number itself. It is a concept of great importance in defining friendly and perfect numbers. Here, we describe a suitable generalization of abundancy index to…
The set of integers which can be written as the sum of four prime cubes has lower density at least $0.009664$. This improves earlier bounds of $0.003125$ by Ren and $0.005776$ by Liu.
We establish an explicit inequality for the number of divisors of an integer $n$. It uses the size of $n$ and its number of distinct prime divisors.
Bounds on the entropy of patterns of sequences generated by independently identically distributed (i.i.d.) sources are derived. A pattern is a sequence of indices that contains all consecutive integer indices in increasing order of first…
Let $F(x)$ be an irreducible polynomial with integer coefficients and degree at least 2. For $x\ge z\ge y\ge 2$, denote by $H_F(x, y, z)$ the number of integers $n\le x$ such that $F(n)$ has at least one divisor $d$ with $y<d\le z$. We…
For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial…
The prime divisors of a polynomial $P$ with integer coefficients are those primes $p$ for which $P(x) \equiv 0 \pmod{p}$ is solvable. Our main result is that the common prime divisors of any several polynomials are exactly the prime…
We report on some statistical regularity properties of greatest common divisors: for large random samples of integers, the number of coprime pairs and the average of the gcd's of those pairs are approximately normal, while the maximum of…
We study odd numbers through a straightforward indexing. We focus in particular on odd prime and composite numbers and their distribution. With a counting argument, we calculate the limit of two sums and compare their convergence rate.
We determine the average number of distinct subsequences in a random binary string, and derive an estimate for the average number of distinct subsequences of a particular length.
We believe we have made progress in the age-old problem of divisibility rules for integers. Universal divisibility rule is introduced for any divisor in any base number system. The divisibility criterion is written down explicitly as a…
In a prime number decomposition of integers in a given set, the occurrence frequencies of prime numbers are shown to satisfy a general forms of Zipf's law.
This paper is devoted to the study of statistical properties of the greatest common divisor and the least common multiple of random samples of positive integers.
Tight bounds on the block entropy of patterns of sequences generated by independent and identically distributed (i.i.d.) sources are derived. A pattern of a sequence is a sequence of integer indices with each index representing the order of…
In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of…