A Survey on Triangular Number, Factorial and Some Associated Numbers

Objectives: The paper aims to present a survey of both time-honored and contemporary studies on triangular number, factorial, relationship between the two, and some other numbers associated with them. Methods: The research is expository in nature. It focuses on expositions regarding the triangular number, its multiplicative analog – the factorial and other numbers related to 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 nobody 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 as to its parity, compositeness, number and sum of positive divisors and other minor characteristics. It was also found that every factoriangular number has a runsum representation of length k, the first term of which is (k −1)! + 1 and the last term is (k −1)! + k . The sequence of factoriangular numbers is a recurring sequence and it has a rational closed-form of exponential generating function. These numbers were also characterized as to when a factoriangular number can be 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 type of number is a novel contribution to the theory of numbers. Surveys, expositions and explorations on existing studies may continue to be a major undertaking in number theory.