{"paper":{"title":"On consecutive abundant numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hui Lv, Yong-Gao Chen","submitted_at":"2016-03-20T03:06:15Z","abstract_excerpt":"A positive integer $n$ is called an abundant number if $\\sigma (n)\\ge 2n$, where $\\sigma (n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\\H os proved that there are two positive constants $c_1$ and $c_2$ such that $c_1\\log\\log\\log x\\le E(x)\\le c_2\\log\\log\\log x$. In this paper, we resolve this old problem by proving that, $E(x)/\\log \\log\\log x$ tends to a limit as $x\\to +\\infty$, and the limit value has an explicit form which is between $3$ and $4$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06176","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":""},"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"}