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Let pod ℓ (n) be the number of ℓ -regular partitions of n with distinct odd parts. In this article, we prove that for any positive integer k, the set of non-negative integers n for which pod ℓ (n) ≡ 0 (mod p k) has density one. We also exhibit several multiplicative identities for pod 3 (n) , pod 5 (n) and pod 7 (n) using the Hecke eigenforms, and some results of Ono, Robins, and Wahl. [ABSTRACT FROM AUTHOR]

We define triangular numerical semigroup as the numerical semigroups generated by all the triangular numbers greater than or equal to a given triangular number and we study the minimal set of generators and the embedding dimension of these numerical semigroups. [ABSTRACT FROM AUTHOR]

In recent work, Z.-H. Sun proposes a large number of conjectural formulas on interrelations between representations by a sum of triangular numbers and representations by ternary or quaternary quadratic forms. In this note, we establish a general statement to illustrate that all these conjectural formulas can be rigorously verified with the aid of modular forms, and as a consequence, we provide a computationally straightforward simple procedure for validating these formulas. [ABSTRACT FROM AUTHOR]

In this section of Resonance, we invite readers to pose questions likely to be raised in a classroom situation. We may suggest strategies for dealing with them, or invite responses, or both. "Classroom" is equally a forum for raising broader issues and sharing personal experiences and viewpoints on matters related to teaching and learning science. [ABSTRACT FROM AUTHOR]

Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique. [ABSTRACT FROM AUTHOR]

The triangular numbers, which are numbers associated with certain arrays of dots, were known to the ancient Greeks and viewed by them with reverence. Though possessing a simple definition, they are astonishingly rich in properties of various kinds, ranging from simple relationships between them and the square numbers to very complex relationships involving partitions, modular forms, etc. - topics which belong to advanced mathematics. They also possess many pretty combinatorial properties. In this expository article we survey a few of these properties. [ABSTRACT FROM AUTHOR]

famous theorem of Euler asserts that there are as many partitions of n into distinct parts as there are partitions into odd parts. We begin by establishing a less well-known companion result, which states that both of these quantities are equal to the number of partitions of n into even parts along with exactly one triangular part. We then introduce the characteristic of a partition, which is determined in a simple way by the placement of odd parts within the list of all parts. This leads to a refinement of the aforementioned result in the form of a new type of partition identity involving characteristic, distinct parts, even parts, and triangular numbers. Our primary purpose is to present a bijective proof of the central instance of this new type of identity, which concerns balanced partitions-partitions in which odd parts occupy as many even as odd positions within the list of all parts. The bijection is accomplished by means of a construction that converts balanced partitions of 2 n into unrestricted partitions of n via a pairing of the squares in the Young tableau. [ABSTRACT FROM AUTHOR]

The article presents information about calculating the perfect median of natural numbers. Brian Hayes created an equation for finding the perfect median of a set of consecutive natural numbers by using two variables. Hayes determined the values of the variables satisfying the equation with the help of a computer program.

In this paper, we demonstrate that 1 is the only integer that is both triangular and a repunit. [ABSTRACT FROM AUTHOR]