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Noah Lebowitz-Lockard

Publications and source records attributed to Noah Lebowitz-Lockard.

12 recordsLinked to original sources
arXiv2025

On pairs of consecutive sequences with the same radicals

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 above, improving a result of Balasubramanian, Shorey, and Waldschmidt. We also bound the number of pairs $(m, n)$ for which $m < n \leq x$ and $m(m + 1) \cdots (m + k - 1))$ and $n(n + 1) \cdots (n + \ell - 1)$ have the same radical and the number of pairs for which $m + i$ and $n + i$ have the same radical for all $i < k$.

math.NT↗
arXiv2025

Non-standard quaternary representations and the Fibonacci numbers

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 $k$ such that $f_4(n+k)=b_4(n)$ for all $n\ge -k$, in contrast to the binary case. Nevertheless, there do exist integers $k$ such that $f_4(n+k)=b_4(n)$ for arbitrarily large intervals of $n$. We generalize these results to any even base $d$. We also study the rate of growth of $b_4(n)$ and show that maximal values of this function correspond to certain Fibonacci numbers.

math.NT↗
arXiv2025

On a theorem of Erdős and Loxton

Let $a(n)$ be the number of partitions of $n$ of the form $a_1 + a_2 + \cdots + a_k$ where $a_{i + 1}$ is a proper divisor of $a_i$ for all $i < k$. Erd{\H o}s and Loxton showed that the sum of $a(n)$ over all $n \leq x$ is asymptotic to a constant multiple of $x^ρ$ where $s = ρ\approx 1.73$ is the unique solution to the equation $ζ(s) = 2$ satisfying $s > 1$. In this note, we provide tight bounds on the value of this constant, though we do not find an exact formula for it. In addition, we write an explicit upper bound for $a(n)$.

math.NT↗
arXiv2024

Runs of integers with constant values of the Carmichael function

In 2023, the first author and Vandehey proved that the largest $k$ for which the string of equalities $λ(n+1)=λ(n+2)=\cdots=λ(n+k)$ holds for some $n\leq x$, where $λ$ is the Carmichael $λ$ function, is bounded above by $O\left((\log x\log\log x)^2\right)$. Their method involved bounding the value of $λ(n + i)$ from below using the prime factorization of $n + i$ for each $i \leq k$. They then used the fact that every $λ(n + i)$ had to satisfy this bound. Here we improve their result by incorporating a reverse counting argument on a result of Baker and Harman on the largest prime factor of a shifted prime.

math.NT↗
arXiv2024

Partitions in which every term but the smallest one is consecutive

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 smallest one. We find formulas relating to the smallest parts of consecutive and almost consecutive partitions. We also find an alternate combinatorial interpretation of the number of almost consecutive partitions of a given integer $n$ and an asymptotic formula for this quantity.

math.CO↗
arXiv2024

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

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 sum for an even number of distinct parts is equal to the number of divisors of $n$. In this article, we initiate the study of the $k$th smallest part of a partition $π$ into distinct parts of any integer $n$, namely $s_k(π)$. Using $s_k(π)$, we generalize the above result for the $k$th smallest parts of partitions for any positive integer $k$ and show its connection with divisor functions for general $k$ and derive interesting special cases. We also study weighted partitions involving $s_k(π)$ with another parameter $z$, which helps us obtain several new combinatorial and analytical results. Finally, we prove sum-of-tails identities associated with the weighted partition function involving $s_k(π)$.

math.NT↗
arXiv2024

On the number of partitions of a number into distinct divisors

Let $p_{\textrm{dsd}} (n)$ be the number of partitions of $n$ into distinct squarefree divisors of $n$. In this note, we find a lower bound for $p_{\textrm{dsd}} (n)$, as well as a sequence of $n$ for which $p_{\textrm{dsd}} (n)$ is unusually large.

math.NT↗
arXiv2022

A note on the number of Egyptian fractions

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 $a/n = 1/m_1 + \dotsc + 1/m_k$ has a solution in positive integers $m_1, \dotsc, m_k$ is bounded above by $n^{1 - 1/2^{k-2} + o(1)}$ as $n$ goes to infinity. The proof is elementary.

math.NT↗
arXiv2021

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

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 values $ϕ(n)$, for $n\le x$, that are coprime to $p$ are asymptotically uniformly distributed among the $p-1$ coprime residue classes modulo $p$, uniformly for $5 \le p \le (\log{x})^A$ (with $A$ fixed but arbitrary). We also show that the values of $s(n)$, for $n$ composite, are uniformly distributed among all $p$ residue classes modulo every $p\le (\log{x})^A$. These appear to be the first results of their kind where the modulus is allowed to grow substantially with $x$.

math.NT↗
arXiv2021

On factorizations into coprime parts

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 factors are coprime. We establish the asymptotic bounds for the sums of $F(n)^β$ and $G(n)^β$ up to $x$ for all real $β$ and the asymptotic bounds for $f(n)^β$ and $g(n)^β$ for all negative $β$.

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arXiv2016

Classifying Subatomic Domains

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 \cite{BC}, the two authors introduce two properties that are slightly weaker than atomicity, which they call "almost atomicity" and "quasiatomicity". In this paper, we classify various subatomic properties and show that they are all distinct.

math.AC↗