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Masser and Shiu \\cite{masser} call the elements of $N_1$ as `sparsely totient numbers' and initiated the study of these numbers. In this article, we establish several results for sparsely totient numbers. First, we show that a squarefree integer divides all sufficiently large sparsely totient numbers and a non-squarefree integer divides infinitely many sparsely totient numbers. Next, we construct explicit infinite families of sparsely totient n"},"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":"1907.09923","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-07-23T14:52:34Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"3b36c06820945b6f5a55654540d9262fd38b3435b49099f619241a01d750c06a","abstract_canon_sha256":"6aab0aa65430d9a3cccf00d3f63c55d89f89f679c3709488a43361ff601b61f7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:49.693006Z","signature_b64":"JW0RkHAsjSimNYt7NwmHZxM03lhpTFTzyJb+/7u1PEpXu1F565KTSdygq5PNfbr9c1u8sPq5jqXo+8L2s5l/Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"897d927d328163abce2b81d419feb5312090ee7bd7eb6aab2739616fa464c956","last_reissued_at":"2026-05-17T23:39:49.692307Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:49.692307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Combinatorial properties of sparsely totient numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Bhuwanesh Rao Patil, Mithun Kumar Das, Pramod Eyyunni","submitted_at":"2019-07-23T14:52:34Z","abstract_excerpt":"Let $N_1(m)=\\max\\{n \\colon \\phi(n) \\leq m\\}$ and $N_1 = \\{N_1(m) \\colon m \\in \\phi(\\mathbb{N})\\}$ where $\\phi(n)$ denotes the Euler's totient function. Masser and Shiu \\cite{masser} call the elements of $N_1$ as `sparsely totient numbers' and initiated the study of these numbers. In this article, we establish several results for sparsely totient numbers. First, we show that a squarefree integer divides all sufficiently large sparsely totient numbers and a non-squarefree integer divides infinitely many sparsely totient numbers. 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