arXiv subjects

Manoj Upreti

Publications and source records attributed to Manoj Upreti.

4 recordsLinked to original sources

arXiv2026

The range and omitted values of a certain sequence involving the partition function

Let $p(n)$ denote the ordinary partition function. Motivated by analogous questions concerning Euler's totient function and its complementary counting function, we study the range of the partition-derived sequence $p(n)-n$. We give combinatorial interpretations of this sequence and investigate both the attained and omitted positive integers. We obtain exact and asymptotic information about the gaps between consecutive attained values and show that the range is remarkably sparse: its counting function has order $(\log x)^2$, and consequently the range has natural density zero. We also extend the discussion to partitions whose Durfee square has side at least a fixed positive integer.

math.CO↗

arXiv2025

A Unified Approach to Calculating Sylvester Sums

In the context of the Frobenius coin problem, given two relatively prime positive integers $a$ and $b$, the set of nonrepresentable numbers consists of positive integers that cannot be expressed as nonnegative integer combination of $a$ and $b$. This work provides a formula for calculating the power sums of all nonrepresentable numbers, also known as the Sylvester sums. Although alternative formulas exist in the literature, our approach is based on an elementary observation. We consider the set of natural numbers from $1$ to $ab - 1$ and compute their total sum in two distinct ways, which leads naturally to the desired Sylvester sums. This method connects an analytic identity with a combinatorial viewpoint, giving a new way to understand these classical quantities. Furthermore, in this paper, we establish a criterion using the division algorithm to determine whether a given positive integer is nonrepresentable.

math.NT↗

arXiv2024

Frobenius numbers associated with Diophantine triples of $x^2+y^2=z^r$ (extended version)

We give an explicit formula for the $p$-Frobenius number of triples associated with Diophantine equations $x^2+y^2=z^r$, that is, the largest positive integer that can only be represented in $p$ ways by combining the three integers of the solutions of Diophantine equations $x^2+y^2=z^r$. When $r=2$, the Frobenius number has already been given.

math.NT↗

arXiv2022

Berkovich-Uncu type Partition Inequalities Concerning Impermissible Sets and Perfect Power Frequencies

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t^{th} power. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t=1. To prove these inequalities, our methods involve constructing injective maps between the relevant sets of partitions. The construction of these maps crucially involves concepts from analysis and calculus, such as explicit maps used to prove countability of N^t, and Jensen's inequality for convex functions, and then merge them with techniques from number theory such as Frobenius numbers, congruence classes, binary numbers and quadratic residues. We also show a connection of our results to colored partitions. Finally, we pose an open problem which seems to be related to power residues and the almost universality of diagonal ternary quadratic forms.

math.CO↗