Related papers: Comparing $n$-dice fixing the sum of the faces
We settle the existence of certain "anti-magic" cubes using combinatorial block designs and graph decompositions to align a handful of small examples.
We present new counterexamples, which provide stronger limitations to sums-differences statements than were previously known. The main idea is to consider non-uniform probability measures.
In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this paper, we present…
We give some new relations for Newman digit sums respectively different modulos and put some problems. In particular, for the odd prime modulos we put an important conjecture.
A quantitative method is described for comparing chess openings. Test openings and baseline openings are run through chess engines under controlled conditions and compared to evaluate the effectiveness of the test openings. The results are…
Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.
We provide numerical procedures for possibly best evaluating the sum of positive series. Our procedures are based on the application of a generalized version of Kummer's test.
This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…
A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to…
Recently, Jha (arXiv:2007.04243, arXiv:2011.11038) has found identities that connect certain sums over the divisors of $n$ to the number of representations of $n$ as a sum of squares and triangular numbers. In this note, we state a…
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special…
We offer several new summation identities involving harmonic numbers, odd harmonic numbers, and Fibonacci numbers. Our results are derived using three different approaches: partial summation, polynomial identities and binomial…
For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all n>30.
All the work made so far on edge-covering a graph by cliques focus on finding the minimum number of cliques that cover the graph. On this paper, we fix the number of cliques that cover a graph by the same number of vertices that the graph…
Inspired by the fact that the sum of the cubes of the first $n$ naturals is equal to the square of their sum, we explore, for each $n$, the Diophantine equation representing all non-trivial sets of $n$ integers with this property. We find…
We create a simple test for distinguishing between sets of primes and random numbers using just the sum-of-digits function. We find that the sum-of-the-digits of prime numbers does not have an equal probability of being odd or even. The…
Computing modular coincidences can show whether a given substitution system, which is supported on a point lattice in R^d, consists of model sets or not. We prove the computatibility of this problem and determine an upper bound for the…
The problem of finding the sum of a polynomial's values is considered. In particular, for any $n\geq 3$, the explicit formula for the sum of the $n$th powers of natural numbers $S_n=\sum_{x=1}^{m}x^{n}$ is proved:…
These informal notes deal with a number of questions related to sums and integrals in analysis.
We determine properties of the set of values of $ [nx] - ([x]/1 + [2x]/2 + \cdots + [nx]/x) $ as $n$ and $x$ vary.