{"paper":{"title":"A sieve problem and its application","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas Weingartner","submitted_at":"2016-05-17T15:19:02Z","abstract_excerpt":"Let $\\theta$ be an arithmetic function and let $\\mathcal{B}$ be the set of positive integers $n=p_1^{\\alpha_1} \\cdots p_k^{\\alpha_k}$, which satisfy $p_{j+1} \\le \\theta ( p_1^{\\alpha_1}\\cdots p_{j}^{\\alpha_{j}})$ for $0\\le j < k$. We show that $\\mathcal{B}$ has a natural density, provide a criterion to determine whether this density is positive, and give various estimates for the counting function of $\\mathcal{B}$. When $\\theta(n)/n$ is non-decreasing, the set $\\mathcal{B}$ coincides with the set of integers $n$ whose divisors $1=d_1< d_2 < \\ldots <d_{\\tau(n)}=n$ satisfy $d_{j+1} \\le \\theta( d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05204","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"}