Port Fillings for Primary Pseudoperfect Numbers

Abstract

Erd\H{o}s asked whether there are infinitely many finite sets of distinct primes $p_1<\cdots<p_k$ and positive integers $m$ such that \begin{equation}\label{eq:erdos-original} \frac1{p_1}+\cdots+\frac1{p_k}=1-\frac1m. \end{equation} This is Erd\H{o}s Problems \#313~\cite{ErdosProblems313}. As recalled below, it is equivalent to the infinitude of primary pseudoperfect numbers. Following Butske, Jaje, and Mayernik~\cite{ButskeJajeMayernik}, a squarefree positive integer $n$ is a \emph{primary pseudoperfect number} if \begin{equation}\label{eq:ppn-def} \frac1n+\sum_{p\mid n}\frac1p=1, \end{equation} where the sum is over the prime divisors of $n$. OEIS A054377~\cite{OEISA054377} records the initial values \[ \begin{array}{c} 2,\ 6,\ 42,\ 1806,\ 47058,\\[2pt] 2214502422,\ 52495396602. \end{array} \] and the eight-prime-factor example \[ \text{\seqsplit{8490421583559688410706771261086}}. \] Butske, Jaje, and Mayernik proved by computation that for each $r\le 8$ there is exactly one primary pseudoperfect number with $r$ distinct prime factors~\cite{ButskeJajeMayernik}. This result gives a useful baseline, but it does not address later layers or the infinitude problem. This paper uses a local language for residual equations. A \emph{port} is a pair $(R,c)$, and a squarefree integer $B$ fills it if \[ \Delta_{R,c}(B):=cB-R\partial(B)=1. \] The corresponding reciprocal form is \[ \sum_{q\mid B}\frac1q+\frac1{RB}=\frac cR. \] The product rule for the arithmetic derivative gives the composition law for ports. This law separates fillings inherited from smaller primary pseudoperfect numbers from fillings that are primitive relative to the fixed residual equation. The unconditional results of the paper are as follows.