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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 $4$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1603.06176","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-20T03:06:15Z","cross_cats_sorted":[],"title_canon_sha256":"f0e13c3d410740692c16a436310a0a8ccb8c2dfc72499c849f07e459bdca9db3","abstract_canon_sha256":"72f488fb53f40e029a6eeab3df9a86c108a9731e51275766d0e077f404399f66"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:50.101121Z","signature_b64":"3JIBLT3aE2souDE3JmyOw8mWmp3h7Oae7L+xxlTqFg4ahyAklknDdgarGY4JZdnf/XF/9K6bwiqPGIy8kRuuAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b30ee39e5e343836188d8930c840b04eaa7c5fe6ff0e17a65be12a64021f28f","last_reissued_at":"2026-05-18T01:18:50.100542Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:50.100542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $n$. 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