{"paper":{"title":"Port Fillings for Primary Pseudoperfect Numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Han Wang","submitted_at":"2026-05-18T12:39:12Z","abstract_excerpt":"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}\n  \\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}\n  \\frac1n+\\sum_{p\\mid n}\\frac1p=1, \\end{eq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21518","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.21518/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}