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\vskip 1cm{\LARGE\bf
Egyptian Fractions with Denominators \\
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from Sequences Closed Under Doubling
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\large
David Eppstein\\
Department of Computer Science\\
University of California, Irvine\\
Irvine, CA 92697\\
USA\\
\href{mailto:eppstein@uci.edu}{\tt eppstein@uci.edu}
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\begin{abstract}
Resolving a conjecture of Zhi-Wei Sun, we prove that every rational number can be represented as a sum of distinct unit fractions whose denominators are practical numbers. The same method applies to allowed denominators that are closed under multiplication by two and include a multiple of every positive integer, including the odious numbers, evil numbers, Hardy-Ramanujan numbers, Jordan-P\'olya numbers, and fibbinary numbers.
\end{abstract}
\section{Introduction}
An \emph{Egyptian fraction} represents a rational number as a sum of distinct unit fractions; we allow $1/1$ as a unit fraction, but no larger integers. Several past works have studied Egyptian fractions with restricted denominators. In the mid-1950s, in connection with the still-open question of the termination of a greedy algorithm for finding representations with odd denominators, Breusch and Stewart showed that all rational numbers with odd denominators have Egyptian fractions with odd denominators~\cite{Bre-AMM-54,Ste-AJM-54}. One of Graham's first publications~\cite{Gra-PJM-64} Egyptian showed that Egyptian fractions with square denominators exist for all rationals in the intervals $[0,\pi^2/6-1)$ and $[1,\pi^2/6)$, and characterized Egyptian fractions with $k$th-power denominators for $k>2$. The last posthumous publication of Erd\H{o}s~\cite{ButErdGra-Int-15} showed that all natural numbers have Egyptian fraction representations whose denominators are the products of three prime numbers.
We find two properties of integer sequences that allow all rational numbers to be represented as Egyptian fractions with denominators from the sequence, or all rational numbers up to the natural limit of such representations, the sum of reciprocals of sequence elements.
\begin{itemize}
\item Sequence $S$ is \emph{closed under doubling} (or \emph{doubling-closed}) when, for all $x\in S$, the quantity $2x$ is also a member of $S$. That is, doubling a sequence element produces another sequence element.
\item Sequence $S$ is \emph{productive} when, for all $x\in\mathbb{Z}^+$, there exists $y\in\mathbb{Z}^+$ such that $xy$ is a member of $S$. That is, every integer has a multiple that is in the sequence.
\end{itemize}
Well-known sequences with both properties include the practical numbers, odious and evil numbers, Hardy-Ramanujan numbers, Jordan-P\'olya numbers, and fibbinary numbers (see Section~\ref{sec:sequences} for details on these sequences), allowing us to find Egyptian fractions using their elements as denominators. Our results positively resolve a conjecture of Sun according to which all rational numbers have Egyptian fractions with practical denominators~\cite{Sun-15}. Two other conjectures of Sun on Egyptian fractions whose denominators are the primes minus one or the primes plus one remain open; the sequences in those conjectures are not closed under doubling.
Our construction method multiplies the numerator of a fraction by a power of two, divides by the denominator, and separately finds the binary representations of the quotient and remainder. This method was used by Stewart to find Egyptian fractions with all denominators even~\cite{Ste-AJM-54} and called the ``binary remainder method'' in our earlier survey on Egyptian fraction construction algorithms~\cite{Epp-MER-95}.
\section{Doubling-closed and productive sequences}
\label{sec:sequences}
We briefly survey some notable integer sequences that are closed under doubling and productive. For these sequences, we also consider the convergence or divergence of the sum
\[ \sum_{x\in S} \frac{1}{x} \]
of reciprocals of elements of a sequence $S$, as this sum (when finite) forms a natural upper bound on Egyptian fractions with denominators in $S$. Some of these sequences are subsequences of others, but for representation by Egyptian fractions this is not redundant, because they will represent different ranges of rational numbers.
\begin{itemize}
\item The practical numbers (\seqnum{A005153}) are the numbers $n$ such that all integers $m\le n$ may be represented as sums of distinct divisors of $n$. For $n$ in this sequence, all rationals $m/n\in(0,1)$ have Egyptian fraction representations with denominators that are divisors of $n$. Several tables of Egyptian fraction expansions based on this principle were given by Fibonacci~\cite{Sig-FLA-02}, but the first explicit definition of the practical numbers was by Srinivasan in 1948~\cite{Sri-CS-48}.
A positive integer $n$ is practical if and only if, for each prime $p$ that divides $n$, the sum of divisors of the $(