NOTE ON ARITHMETIC PROPERTIES OF KUMMER'S FUNCTION

Arithmetic properties of the Ramanujan function

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisor P(τ(n)) and the number of distinct prime divisors ω(τ(n)) of τ(n) for various sequences of n. In particular, we show that P(τ(n)) ≥ (logn)33/31+o(1) for infinitely many n, and P(τ(p)τ(p2)τ(p3)) > (1+o(1)) log log p log log log p loglog log log p for every prime p with τ(p) 6= 0.

Arithmetic Properties of a Restricted Bipartition Function

A bipartition of n is an ordered pair of partitions (λ, μ) such that the sum of all of the parts equals n. In this article, we concentrate on the function c5(n), which counts the number of bipartitions (λ, μ) of n subject to the restriction that each part of μ is divisible by 5. We explicitly establish four Ramanujan type congruences and several infinite families of congruences for c5(n) modulo 3.

A Note on the First Geometric-Arithmetic Index of Hexagonal Systems and Phenylenes

The first geometric-arithmetic index was introduced in the chemical theory as the summation of 2 du dv /(du  dv ) overall edges of the graph, where du stand for the degree of the vertex u. In this paper we give the expressions for computing the first geometric-arithmetic index of hexagonal systems and phenylenes and present new method for describing hexagonal system by corresponding a simple g...

A Note on Arithmetic Constraint Propagation

We consider the resolution by constraint programming of large problems, i.e. involving millions of constraints, which mainly imply arithmetic constraints, like shortest path problems or other related problems. We show that a simple constraint programming model is not competitive with dedicated algorithms (or dedicated constraints). This mainly comes from the propagation mechanism, i.e. the orde...

A Note on Rational Arithmetic Functions