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Similarly, ${\\cal F}$ denotes the family of all finite nonempty $T\\subseteq{\\mathbb Q}^+$, and ${\\cal F}(Y) := {\\cal F}\\cap\\{T:T\\subseteq Y\\}$ where ${\\mathbb Q}^+$ is the set of all positive rationals and $Y\\subseteq{\\mathbb Q}^+$.\n  This paper treats the functions $\\sigma:{\\cal E}\\rightarrow{\\mathbb Q}^+$ given by $\\sigma:S\\mapsto\\sigma S :=\\sum\\{1/x:x\\in S\\}$, the function $\\delta:{\\cal E}\\rightarrow{\\mathbb N}$ defined"},"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":"1702.01316","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-02-04T17:56:04Z","cross_cats_sorted":[],"title_canon_sha256":"4f5a274ae9b5e200d6fd8a95cd466d3a9d35685fa2d75b99e88f4a431adb899e","abstract_canon_sha256":"ff18084d113b7d60768c4abfaded79dfcf6e2b59471a87256e8c813accba2388"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:44.557663Z","signature_b64":"9Xic0uqjDAPYIPOSauCZhPA+GFndzFAhq9EUuR4Pyq5+q55nphsrGpxEV71ErGR6M5QL0ngDqO7Gey0YrDN6Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d193ec21b7fa8332198e540b7648a4242fd6b876551d8fa29a9d3079fdeb99a5","last_reissued_at":"2026-05-17T23:53:44.557251Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:44.557251Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sums of finitely many distinct rationals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Hobby, Donald Silberger, Sylvia Silberger","submitted_at":"2017-02-04T17:56:04Z","abstract_excerpt":"${\\cal E}$ denotes the family of all finite nonempty $S\\subseteq{\\mathbb N}:=\\{1,2,\\ldots\\}$, and ${\\cal E}(X):={\\cal E}\\cap\\{S:S\\subseteq X\\}$ when $X\\subseteq{\\mathbb N}$. Similarly, ${\\cal F}$ denotes the family of all finite nonempty $T\\subseteq{\\mathbb Q}^+$, and ${\\cal F}(Y) := {\\cal F}\\cap\\{T:T\\subseteq Y\\}$ where ${\\mathbb Q}^+$ is the set of all positive rationals and $Y\\subseteq{\\mathbb Q}^+$.\n  This paper treats the functions $\\sigma:{\\cal E}\\rightarrow{\\mathbb Q}^+$ given by $\\sigma:S\\mapsto\\sigma S :=\\sum\\{1/x:x\\in S\\}$, the function $\\delta:{\\cal E}\\rightarrow{\\mathbb N}$ defined"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01316","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1702.01316","created_at":"2026-05-17T23:53:44.557321+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.01316v2","created_at":"2026-05-17T23:53:44.557321+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.01316","created_at":"2026-05-17T23:53:44.557321+00:00"},{"alias_kind":"pith_short_12","alias_value":"2GJ6YINX7KBT","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_16","alias_value":"2GJ6YINX7KBTEGMO","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_8","alias_value":"2GJ6YINX","created_at":"2026-05-18T12:30:55.937587+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2GJ6YINX7KBTEGMOKQFXMSFEEQ","json":"https://pith.science/pith/2GJ6YINX7KBTEGMOKQFXMSFEEQ.json","graph_json":"https://pith.science/api/pith-number/2GJ6YINX7KBTEGMOKQFXMSFEEQ/graph.json","events_json":"https://pith.science/api/pith-number/2GJ6YINX7KBTEGMOKQFXMSFEEQ/events.json","paper":"https://pith.science/paper/2GJ6YINX"},"agent_actions":{"view_html":"https://pith.science/pith/2GJ6YINX7KBTEGMOKQFXMSFEEQ","download_json":"https://pith.science/pith/2GJ6YINX7KBTEGMOKQFXMSFEEQ.json","view_paper":"https://pith.science/paper/2GJ6YINX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.01316&json=true","fetch_graph":"https://pith.science/api/pith-number/2GJ6YINX7KBTEGMOKQFXMSFEEQ/graph.json","fetch_events":"https://pith.science/api/pith-number/2GJ6YINX7KBTEGMOKQFXMSFEEQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2GJ6YINX7KBTEGMOKQFXMSFEEQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2GJ6YINX7KBTEGMOKQFXMSFEEQ/action/storage_attestation","attest_author":"https://pith.science/pith/2GJ6YINX7KBTEGMOKQFXMSFEEQ/action/author_attestation","sign_citation":"https://pith.science/pith/2GJ6YINX7KBTEGMOKQFXMSFEEQ/action/citation_signature","submit_replication":"https://pith.science/pith/2GJ6YINX7KBTEGMOKQFXMSFEEQ/action/replication_record"}},"created_at":"2026-05-17T23:53:44.557321+00:00","updated_at":"2026-05-17T23:53:44.557321+00:00"}