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We also prove that there are infinitely many $n$ for which \\[\n  \\max \\{ {\\rm sfp}(n), {\\rm sfp}(n+1), {\\rm sfp}(n+2) \\} < n^{1/3}. \\]"},"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":"1502.00605","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-02-02T20:15:37Z","cross_cats_sorted":[],"title_canon_sha256":"8fa59c4df5a6eb71825a3a468b9d328e38256aa4b48be908d5dcfa7248775064","abstract_canon_sha256":"653f49c62d21be65ebcffbf927fa63bfcafdb97622f68138754cffe068baebf5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:25:35.187526Z","signature_b64":"woWyTtWS1JY/ldWaCeJl0lTZ2cXJeUbKt0e6AAXA5pbNWaDETsvTiehKMahRoR1TrfrhFaCs0oHMN6gG4wsfDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf8f80ba4f005a8548795e3875e3bda6ba82cf7dc37d61a44b8f210e81123577","last_reissued_at":"2026-05-18T00:25:35.186905Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:25:35.186905Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Three consecutive almost squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jeremy Rouse, Yilin Yang","submitted_at":"2015-02-02T20:15:37Z","abstract_excerpt":"Given a positive integer $n$, we let ${\\rm sfp}(n)$ denote the squarefree part of $n$. 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