Related papers: On a sequence related to the Josephus problem

In this paper, we investigate properties of the fixed point sequence of the Josephus function $J_3$. First, we establish a connection between this sequence and the Chinese Remainder Theorem. Next, we identify a clear numerical pattern for…

Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the…

The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…

This is an introduction to small divisors problems. The material treated in this book was brought together for a PhD course I tought at the University of Pisa in the spring of 1999. Here is a Table of Contents: Part I One Dimensional Small…

We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of…

The question of the convergence of generalized formal power series (with complex power exponents) solutions of $q$-difference equations is studied in the situation where the small divisors phenomenon arises; a sufficient condition of…

In this paper, we study various classes of partition functions such as those related to the parity of the number of parts, to differences of partition numbers, and to partitions with a repeated smallest part. We establish identities…

We reduce the principal problem of Additive Number Theory of whether an infinite sequence of integers constitutes a finite basis for the integers to a Diophantine problem involving the difference set of the sequence, by proving a formula…

For positive integers $s$ and $L \geq 3$, Berkovich and Uncu (Ann. Comb. $23$ ($2019$) $263$--$284$) conjectured an inequality between the sizes of two closely related sets of partitions whose parts lie in the interval $\{s, \ldots, L+s\}$.…

In the first part we associate a periodic sequence to a partition and study the connection the distribution of elements of uniform limit of the sequences. Then some facts of statistical independence of these limits are proved

In this paper, we provide a comprehensive solution to the open problem regarding the existence of a recurrence formula for computing fixed points of the Josephus function precisely when the reduction constant is three. Incorporating this…

We continue work started in [1] concerning integer sequences q(n), n in N, defined by q(n) = q(n-q(n-1)) + f(n), with q(1) = 1. Here, f(n), with f(1) = 0, is a given sequence. We define F as the set of semi-infinite sequence f such that the…

The set of hook lengths of an integer partition $\lambda$ is the complement of some numerical semigroup $S$. There has been recent interest in studying the number of partitions with a given set of hook lengths. Very little is known about…

The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…

We investigate the partitioning of partial orders into a minimal number of heapable subsets. We prove a characterization result reminiscent of the proof of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for computing…

This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a beautiful structure. It is interesting to note that…

This paper describes a simple method for estimating lower bounds on the number of classes of equivalence for a special kind of integer sequences, called division sequences. The method is based on adding group structure to classes of…

Let $1\leq a<q$ be a pair of small integers such that $\gcd(a,q)=1$ and let $x>1$ be a large number. This note discusses the existence of a short sequence of primes $p\equiv a\bmod q$ between two squares $x^2$ and $(x+1)^2$.

Given a multiplicative function $f$, we let $S(x,f)=\sum_{n\leq x}f(n)$ be the associated partial sum. In this note, we show that lower bounds on partial sums of divisor-bounded functions result in lower bounds on the partial sums…