{"paper":{"title":"Counting Square-Free Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.NT","authors_text":"Jakub Pawlewicz","submitted_at":"2011-07-25T10:55:58Z","abstract_excerpt":"The main topic of this contribution is the problem of counting square-free numbers not exceeding $n$. Before this work we were able to do it in time (Comparing to the Big-O notation, Soft-O ($\\softO$) ignores logarithmic factors) $\\softO(\\sqrt{n})$. Here, the algorithm with time complexity $\\softO(n^{2/5})$ and with memory complexity $\\softO(n^{1/5})$ is presented. Additionally, a parallel version is shown, which achieves full scalability.\n  As of now the highest computed value was for $n=10^{17}$. Using our implementation we were able to calculate the value for $n=10^{36}$ on a cluster."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.4890","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"}