Giuseppe Melfi

On a question about sum-free sequences

Abstract An increasing sequence of positive integers {n1, n2, …} is called a sum-free sequence if every term is never a sum of distinct smaller terms. We prove that there exist sum-free sequences {nk} with polynomial growth and such that limk→∞ nk+1/nk = 1.

A Different Approach for the Analysis of Web Access Logs

The development of Internet-based business has pointed out the importance of the personalisation and optimisation of Web sites. For this purpose the study of users behaviours are of great importance. In this paper we present a solution to the problem of identification of dense clusters in the analysis of Web Access Logs. We consider a modification of an algorithm recently proposed in social network analysis. This approach is illustrated by analysing a log-file of a web portal.

On simultaneous binary expansions of n and n2

Abstract A new family of increasing sequences of positive integers is proposed. The integers n for which the sum of binary digits is equal to the sum of binary digits of n 2 are an example of sequence of this family and this sequence is more accurately studied. Some structure and asymptotic properties are proved and a conjecture about its counting function is discussed.

Random number generators and rare events in the continued fraction of π

Failure of pseudo-random number generators in producing reliable random numbers as described by Knuth (Knuth, D.E., 1981, The Art of Computer Programming, Vol. 2, Addison-Wesley) gave birth to a new generation of random number generators such as billions decimals of π. To show that these decimals satisfy all criterion of being random, Bailey and Crandall (Bailey, D.B. and Crandall, R.E., 2003, Random generators and normal numbers, to appear in Experimental Mathematics) provided a proof toward the normality of π. In this article, we try to show similar results by considering the continued fraction of π, which appears random as opposed to other supposed normal numbers whose continued fractions are not random at all. For this purpose, we analyze the continued fraction of π and discuss the randomness of its partial quotients. Some statistical tests are performed to check whether partial quotients follow the Khinchin distribution. Finally, we discuss rare elements in the continued fraction of π.