应用与计算数学杂志

A Survey on Triangular Number and Factorial Number

Abstract

Castillo Jack

Objectives:The paper aims to present a survey of both time-honored and contemporary studies on triangular number, factorial, relationship between the 2, and a few other numbers related to them.

Methods: The research is expository in nature. It focuses on expositions regarding the triangular number, its multiplicative analog – the factorial and other numbers associated with them.

Findings: Much had been studied about triangular numbers, factorials and other numbers involving sums of triangular numbers or sums of factorials. However, it seems that no-one had explored the properties of the sums of corresponding factorials and triangular numbers. Hence, explorations on these integers, called factoriangular numbers, were conducted. Series of experimental mathematics resulted to the characterization of factoriangular numbers on its parity, compositeness, number and sum of positive divisors and other minor characteristics. The sequence of factoriangular numbers may be a recurring sequence and it's a rational closed-form of exponential generating function. These numbers were also characterized on when a factoriangular number are often expressed as a sum of two triangular numbers and/or as a sum of two squares.

Application/ Improvement: The introduction of factoriangular number and expositions on this sort of number may be a novel contribution to the idea of numbers. Surveys, expositions and explorations on existing studies may still be a serious undertaking in number theory.