Zobrazeno 1 - 10
of 31
pro vyhledávání: '"Zelinsky, Joshua"'
We investigate the Dots and Boxes game, also known as ``Strings and Coins,'' for certain specific families of graphs. These include complete graphs, wheel graphs, and friendship graphs.
Externí odkaz:
http://arxiv.org/abs/2407.15198
Autor:
Zelinsky, Joshua
Write $T(n)$ as the sum of the reciprocals of the primes which divide $n$. Write $H(n) = \prod_{p|n}p/(p-1)$ where the product is over the prime divisors of $n$. We prove new bounds for $T(n)$ and $H(n)$ in terms of the smallest prime factor of $n$,
Externí odkaz:
http://arxiv.org/abs/2402.14234
Autor:
McCormack, Tim, Zelinsky, Joshua
We use the weighted version of the arithmetic-mean-geometric-mean inequality to motivate new results about Zaremba's function, $z(n) = \sum_{d|n} \frac{\log d}{d}$. We investigate record-setting values for $z(n)$ and the related function $v(n) = \fra
Externí odkaz:
http://arxiv.org/abs/2312.11661
Autor:
Gallardo, Luis H., Zelinsky, Joshua
In this note, we fix a gap in a proof of the first author that 28 is the only even perfect number which is the sum of two perfect cubes. We also discuss the situation for higher powers.
Externí odkaz:
http://arxiv.org/abs/2310.07077
Let $\sigma(n)$ be the sum of the positive divisors of $n$. A number $n$ is said to be 2-near perfect if $\sigma(n) = 2n +d_1 +d_2 $, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We give a complete description of those $n$ which are 2
Externí odkaz:
http://arxiv.org/abs/2310.01305
Autor:
Zelinsky, Joshua
Define $||n||$ to be the \emph{complexity} of $n$, which is the smallest number of $1$s needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $||n|| \geq 3\log_3 n$ for all $n$. Richard Guy note
Externí odkaz:
http://arxiv.org/abs/2211.02995
Autor:
Benson-Tilsen, Noam, Brock, Samuel, Faunce, Brandon, Kumar, Monish, Stein, Noah Dokko, Zelinsky, Joshua
Publikováno v:
Involve 16 (2023) 765-781
Given a graph $G$, a \textit{$k$-total difference labeling} of the graph is a total labeling $f$ from the set of edges and vertices to the set $\{1, 2, \cdots k\}$ satisfying that for any edge $\{u,v\}$, $f(\{u,v\})=|f(u)-f(v)|$. If $G$ is a graph, t
Externí odkaz:
http://arxiv.org/abs/2107.11706
Autor:
Zelinsky, Joshua
Let $n$ be a primitive non-deficient number where $n=p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1, p_2 \cdots p_k$ are distinct primes. We prove that there exists an $i$ such that $$p_i^{a_i+1} < 2k(p_1p_2p_3\cdots p_k).$$ We conjecture that in fa
Externí odkaz:
http://arxiv.org/abs/2005.12115
Autor:
Zelinsky, Joshua
Let $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second smallest, th
Externí odkaz:
http://arxiv.org/abs/2005.12118
Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as the closel
Externí odkaz:
http://arxiv.org/abs/1908.09420