Volume 22, 2016, Number 1

Let p_n be the n-th prime number. The following limit is well-known

Let k be a fixed but arbitrary nonnegative integer. In this note we prove the more general limit

We point out that an inequality published recently in [1] is a particular case of a general result from [4]. By another method, a refinement is offered, too. Related inequalities are also proved.

In this paper, let p denote a prime. We shall consider sums of the type Φ (x,y;f)= Σ_{n≤p|n ⇒ p > y} f(n) and ψ (x,y;f)=Σ_{n≤p|n ⇒ p < y} f(n) for certain kinds of arithmetical functions f and prove some identities for Φ and ψ which are analogous to the ‘so-called’ Buchstab identity. As an application, we will prove some formulas for square-free integers.

In this paper, we compute the generalized bivariate Fibonacci and Lucas p-polynomials by using inverse of various triangular matrices. In addition, in each calculation, instead of obtaining a type of sequence only, we are able to determine successive n terms of the two types of polynomial sequences simultaneously.

For prime p and 1 ≤ a, b, c < p let V be the algebraic set of the congruence ax − by ≡ c (mod p) in the plane. For an arbitrary box of size B we obtain a necessary and a sufficient conditions on the size B in order for the box to meet V. For arbitrary subsets S, T of Z_p we also obtain a necessary and a sufficient conditions on the cardinalities of S, T so that S + T = Z_p.

Various characteristics of the ordinary Fibonacci and Lucas sequences, many known for centuries, are shown to be common to generalized sequences related to the Golden Ratio. Periodicity properties are also investigated.

In this paper, we consider a generalized m-excedance statistic on set partitions which is analogous to the usual excedance statistic on permutations when m = 0. We study it from a probabilistic perspective in which set partitions are regarded as geometrically distributed words satisfying a so-called restricted growth property. We derive a general set of recurrences satisfied by the relevant generating functions and, in the m = 0 and m = 1 cases, find an explicit formula for the distribution of the statistic recording the number of m-excedances in partitions having a fixed number of blocks. When m = 0, one can also determine the joint distribution for the number of excedances with the major index statistic on set partitions. Further recurrences may be given in this case as well as formulas for the mean number of excedances and the variance.

An extension of the best known Pascal’s triangle is introduced and an explicit formula for its members is given.

We discuss families of triples of graphs whose Hosoya indices are primitive Pythagorean triples. Hosoya gave a method to construct such families of caterpillars, i.e., trees whose vertices are within distance 1 of a central path. He also pointed out a common structure to the families. In this paper, we show the uniqueness of the common structure.