Related papers: On Amicable Numbers
In this paper, we give some interesting identities of higher-order Bernoulli, Frobenius-Euler and Euler polynomials arising from umbral calculus. From our method of this paper, we can derive many interesting identities of special…
Translation from the Latin of Euler's "Demonstratio theorematis circa ordinem in summis divisorum observatum" (1760). E244 in the Enestroem index. In his previous paper E243, Euler stated the pentagonal number theorem and assuming it proved…
The aim of this paper is to study degenerate Eulerian polynomials and degenerate Eulerian numbers, respectively as degenerate versions of the Eulerian polynomials and the Eulerian numbers, and to derive some of their properties.…
The paper introduces the notions of an elliptic pair, an elliptic cycle and an elliptic list over a square free positive integer d. These concepts are related to the notions of amicable pairs of primes and aliquot cycles that were…
The study of degenerate versions of certain special polynomials and numbers, which was initiated by Carlitz's work on degenerate Euler and degenerate Bernoulli polynomials, has recently seen renewed interest among mathematicians. The aim of…
In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is connected with an autonomous first-order differential equation, solutions of which are used to obtain integral…
We study average case approximation of Euler and Wiener integrated processes of d variables which are almost surely r_k-times continuously differentiable with respect to the k-th variable. Let n(h,d) denote the minimal number of continuous…
Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n>1 and d>1.
A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of an algebraic integer are also discussed.…
We give thirty-two diverse proofs of a small mathematical gem--the fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many…
Certain generalization of Euler numbers was defined in 1935 by Lehmer using cubic roots of unity, as a natural generalization of Bernoulli and Euler numbers. In this paper, Lehmer's generalized Euler numbers are studied to give certain…
We give several families of polynomials which are related by Eulerian summation operators. They satisfy interesting combinatorial properties like being integer-valued at integral points. This involves nearby-symmetries and a recursion for…
We consider a classial case of irrational integrals containing a square root of a quadratic polynomial. It is well known that they can be expressed in terms of elementary functions by one of three Euler's substitutions. It is less known…
In order to study signed Eulerian numbers, we introduce permutations of a particular type, called parity-alternate permutations, because they take even and odd entries alternately. The objective of this paper is twofold. The first is to…
The numbers of even and odd permutations with a given ascent number are investigated using an operator that was previously introduced by the author. Their difference is called a signed Eulerian number. By means of the operator the…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
In this paper Euler shows that there are no additional square idoneal numbers aside from 1, 4, 9, 16, and 25.
Translation from the Latin original "Utrum hic numerus 1000009 sit primus necne inquiritur" (1778). E699 in the Enestrom index. The idea of this paper is that if some number is a sum of two squares in two ways, then some other smaller…
We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that \[ A,\ A^A,\ A^{A^A},\ \ldots \]…
An interplay between the Lambert series and Euler's Pentagonal Number Theorem gives an Euler-type recurrence relation for any given arithmetical function. As consequences of this, we present Euler-type recurrence relations for some…