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We exhibit a positive function $\\theta(c)$ with the property that the largest prime factor of $\\fl{n^c}$ exceeds $n^{\\theta(c)-\\eps}$ infinitely often. For $c\\in(1,\\tfrac{149}{87})$ we show that the counting function of natural numbers $n\\le x$ for which $\\fl{n^c}$ is squarefree satisfies the expected asymptotic formula. For $c\\in(1,\\tfrac{147}{145})$ we show that there are infinitely many Carmichael numbers composed entirely of primes of the form $p=\\fl{n^c}$."},"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":"1203.5884","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-03-27T07:31:35Z","cross_cats_sorted":[],"title_canon_sha256":"1550179c936ebef12a6ff4d5373c7e6f3f12cede42229193631d0349673701c3","abstract_canon_sha256":"2c3989237caf95f9c6cb3ebad6ae3b674eb701e21d8571853a66e6fd6103f20a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:13.161396Z","signature_b64":"KF1jKC2rBUDw4SQYYiTmgcFpM77ETYFjy3xCQOCrJ+ykMv5myKlnzKnHsNtEu8e07JRAfm1wm81Pci3Cx9G7CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e31157c43da51d99dfe1c0f28f222c1b4bd9dfde700aaf2e4f8652c987831e7","last_reissued_at":"2026-05-18T03:59:13.160800Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:13.160800Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Piatetski-Shapiro sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andreas J. 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For $c\\in(1,\\tfrac{147}{145})$ we show that there are infinitely many Carmichael numbers composed entirely of primes of the form $p=\\fl{n^c}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.5884","kind":"arxiv","version":1},"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":"1203.5884","created_at":"2026-05-18T03:59:13.160905+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.5884v1","created_at":"2026-05-18T03:59:13.160905+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.5884","created_at":"2026-05-18T03:59:13.160905+00:00"},{"alias_kind":"pith_short_12","alias_value":"LYYRK7CD3JI5","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"LYYRK7CD3JI5THP6","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"LYYRK7CD","created_at":"2026-05-18T12:27:14.488303+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/LYYRK7CD3JI5THP6DQHSR4RCYG","json":"https://pith.science/pith/LYYRK7CD3JI5THP6DQHSR4RCYG.json","graph_json":"https://pith.science/api/pith-number/LYYRK7CD3JI5THP6DQHSR4RCYG/graph.json","events_json":"https://pith.science/api/pith-number/LYYRK7CD3JI5THP6DQHSR4RCYG/events.json","paper":"https://pith.science/paper/LYYRK7CD"},"agent_actions":{"view_html":"https://pith.science/pith/LYYRK7CD3JI5THP6DQHSR4RCYG","download_json":"https://pith.science/pith/LYYRK7CD3JI5THP6DQHSR4RCYG.json","view_paper":"https://pith.science/paper/LYYRK7CD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.5884&json=true","fetch_graph":"https://pith.science/api/pith-number/LYYRK7CD3JI5THP6DQHSR4RCYG/graph.json","fetch_events":"https://pith.science/api/pith-number/LYYRK7CD3JI5THP6DQHSR4RCYG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LYYRK7CD3JI5THP6DQHSR4RCYG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LYYRK7CD3JI5THP6DQHSR4RCYG/action/storage_attestation","attest_author":"https://pith.science/pith/LYYRK7CD3JI5THP6DQHSR4RCYG/action/author_attestation","sign_citation":"https://pith.science/pith/LYYRK7CD3JI5THP6DQHSR4RCYG/action/citation_signature","submit_replication":"https://pith.science/pith/LYYRK7CD3JI5THP6DQHSR4RCYG/action/replication_record"}},"created_at":"2026-05-18T03:59:13.160905+00:00","updated_at":"2026-05-18T03:59:13.160905+00:00"}