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We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$ becomes the arithmetic function introduced previously by Farhi. Farhi proved that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved Farhi's period $k!$ to ${\\rm lcm}(1,2,...,k)$ and conjectured that $\\frac{{\\rm lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$. Recently, Farhi and Kane proved th"},"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":"0903.0530","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-03-03T13:31:38Z","cross_cats_sorted":[],"title_canon_sha256":"52577f4140e3b4d23918e72df8a5f7ed26af30821a072a21d6cf5d7bf5925cbd","abstract_canon_sha256":"6838bff378a307818ac8c1cc1cca061c591146c9ca124ed6c8f9a27f6637a14e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:32.427641Z","signature_b64":"g5CwLokxL3dWHtLwmYWXg1tsz7xfP1qc6fkn5xtUknQYAYNls4SnaXkXBc6uAxdxMdopqF/lOIAOIfv0kKf+DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ba0d15479fc0be489f27f35f0c0bbd1963c512dc576c910bed63e7655b87d38a","last_reissued_at":"2026-05-18T04:16:32.427195Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:32.427195Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The least common multiple of consecutive arithmetic progression terms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Guoyou Qian, Shaofang Hong","submitted_at":"2009-03-03T13:31:38Z","abstract_excerpt":"Let $k\\ge 0,a\\ge 1$ and $b\\ge 0$ be integers. We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and $b=0$, then $g_{k,a,b}$ becomes the arithmetic function introduced previously by Farhi. Farhi proved that $g_{k,1,0}$ is periodic and that $k!$ is a period. Hong and Yang improved Farhi's period $k!$ to ${\\rm lcm}(1,2,...,k)$ and conjectured that $\\frac{{\\rm lcm}(1,2,...,k,k+1)}{k+1}$ divides the smallest period of $g_{k,1,0}$. 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