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For any $k$-tuple $\\vec{s}=(s_1, ..., s_k)$ of positive integers, we define $$H_{k,f}(\\vec{s}, n):=\\sum\\limits_{1\\leq i_{1}<\\cdots<i_{k}\\le n} \\prod\\limits_{j=1}^{k}\\frac{1}{f(i_{j})^{s_j}}$$ and $$H_{k,f}^*(\\vec{s}, n):=\\sum\\limits_{1\\leq i_{1}\\leq \\cdots\\leq i_{k}\\leq n} \\prod\\limits_{j=1}^{k}\\frac{1}{f(i_{j})^{s_j}}.$$ If all $s_j$ are 1, then let $H_{k,f}(\\vec{s}, n):=H_{k,f}(n)$ and $H_{k,f}^*(\\vec{s}, n):=H_{k,f}^*(n)$. Hong an"},"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":"1703.07263","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-03-21T15:16:43Z","cross_cats_sorted":[],"title_canon_sha256":"bf1fc47f65b751ad8ff0734acbebb119596f08d8b620c13227b897700ca500d2","abstract_canon_sha256":"79ba3efc88884856053fa9f1ad7e3b5615bfe517be9dfd7905d44629aad04e3a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:00.582746Z","signature_b64":"JRUdhV1AbIa6x/UNkG2VEnLc72zzQZBv5W35DhgTn7yAes09tY7BDYF6yJoVbKTWTnvDzMJ5P/4jggIqgyCuAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8523dd1bab2d21dc5842f85f7d9b2b29b840c9ab1423286aac5548b7df07c902","last_reissued_at":"2026-05-18T00:12:00.582128Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:00.582128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multiple reciprocal sums and multiple reciprocal star sums of polynomials are almost never integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liping Yang, Min Qiu, Qiuyu Yin, Shaofang Hong","submitted_at":"2017-03-21T15:16:43Z","abstract_excerpt":"Let $n$ and $k$ be integers such that $1\\le k\\le n$ and $f(x)$ be a nonzero polynomial of integer coefficients such that $f(m)\\ne 0$ for any positive integer $m$. For any $k$-tuple $\\vec{s}=(s_1, ..., s_k)$ of positive integers, we define $$H_{k,f}(\\vec{s}, n):=\\sum\\limits_{1\\leq i_{1}<\\cdots<i_{k}\\le n} \\prod\\limits_{j=1}^{k}\\frac{1}{f(i_{j})^{s_j}}$$ and $$H_{k,f}^*(\\vec{s}, n):=\\sum\\limits_{1\\leq i_{1}\\leq \\cdots\\leq i_{k}\\leq n} \\prod\\limits_{j=1}^{k}\\frac{1}{f(i_{j})^{s_j}}.$$ If all $s_j$ are 1, then let $H_{k,f}(\\vec{s}, n):=H_{k,f}(n)$ and $H_{k,f}^*(\\vec{s}, n):=H_{k,f}^*(n)$. 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