Noah Lebowitz-Lockard

activity

20162025

most citedA note on the number of Egyptian fractions

2 citations · 3 across the 9 of their papers we have counts for

collaborators

12 papers

math.NT2025

On pairs of consecutive sequences with the same radicals

Noah Lebowitz-Lockard

Let $(m, n, k)$ be a tuple of integers with the property that if $i \leq k$ , then $m + i$ and $n + i$ have the same radical. Using a result on the abc Conjecture, we bound $k$ from…

math.NT2025

Non-standard quaternary representations and the Fibonacci numbers

Katie Anders, Madeline L. Dawsey, Rajat Gupta +2

Let $f_{4} (n)$ be the number of hyperquaternary representations of $n$ and $b_{4} (n)$ be the number of balanced quaternary representations of $n$ . We show that there is no integer $k$ …

math.NT2025

On a theorem of Erdős and Loxton

Noah Lebowitz-Lockard

Let $a (n)$ be the number of partitions of $n$ of the form $a_{1} + a_{2} + \dots + a_{k}$ where $a_{i + 1}$ is a proper divisor of $a_{i}$ for all $i < k$ . Erd{\H o}s and Loxton showed th…

math.CO2024

Partitions in which every term but the smallest one is consecutive

Rajat Gupta, Noah Lebowitz-Lockard

In this article, we introduce the notion of almost consecutive partitions. A partition is almost consecutive if every term is consecutive, with the possible exception of the smalle…

math.NT2024

On the $k$ th smallest part of a partition into distinct parts

Rajat Gupta, Noah Lebowitz-Lockard, Joseph Vandehey

A classic theorem of Uchimura states that the difference between the sum of the smallest parts of the partitions of $n$ into an odd number of distinct parts and the corresponding s…

math.NT2024

On the number of partitions of a number into distinct divisors

Noah Lebowitz-Lockard, Joseph Vandehey

Let $p_{dsd} (n)$ be the number of partitions of $n$ into distinct squarefree divisors of $n$ . In this note, we find a lower bound for $p_{dsd} (n)$ , as well as a…

math.NT2024

Runs of integers with constant values of the Carmichael function

Noah Lebowitz-Lockard, J. C. Saunders

In 2023, the first author and Vandehey proved that the largest $k$ for which the string of equalities $λ (n + 1) = λ (n + 2) = \dots = λ (n + k)$ holds for some $n \leq x$ , where $λ$ is the Carmi…

math.NT2022★ 2 cited

A note on the number of Egyptian fractions

Noah Lebowitz-Lockard, Victor Souza

Refining an estimate of Croot, Dobbs, Friedlander, Hetzel and Pappalardi, we show that for all $k \geq 2$ , the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/…

math.NT2021

Distribution mod $p$ of Euler's totient and the sum of proper divisors

Noah Lebowitz-Lockard, Paul Pollack, Akash Singha Roy

We consider the distribution in residue classes modulo primes $p$ of Euler's totient function $ϕ (n)$ and the sum-of-proper-divisors function $s (n) := σ (n) - n$ . We prove that the value…

math.NT2021

On factorizations into coprime parts

Matthew Just, Noah Lebowitz-Lockard

Let $f (n)$ and $g (n)$ be the number of unordered and ordered factorizations of $n$ into integers larger than one. Let $F (n)$ and $G (n)$ have the additional restriction that the fac…

math.NT2018

Irreducible quadratic polynomials and Euler's function

Noah Lebowitz-Lockard

Let $P (x)$ be an irreducible quadratic polynomial in $Z [x]$ . We show that for almost all $n$ , $P (n)$ does not lie in the range of Euler's totient function.

math.AC2016★ 1 cited

Classifying Subatomic Domains

Noah Lebowitz-Lockard

In general, ring theory is focused on atomic rings, i.e. rings in which every element has some factorization into irreducible elements. In a recent paper of Boynton and Coykendall…

On pairs of consecutive sequences with the same radicals

Non-standard quaternary representations and the Fibonacci numbers

On a theorem of Erdős and Loxton

Partitions in which every term but the smallest one is consecutive

On the kth smallest part of a partition into distinct parts

On the number of partitions of a number into distinct divisors

Runs of integers with constant values of the Carmichael function

A note on the number of Egyptian fractions

Distribution mod p of Euler's totient and the sum of proper divisors

On factorizations into coprime parts

Irreducible quadratic polynomials and Euler's function

Classifying Subatomic Domains

On the $k$ th smallest part of a partition into distinct parts

Distribution mod $p$ of Euler's totient and the sum of proper divisors