{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:VDTNGCJR6ENKZT6UYJ747DEP2P","short_pith_number":"pith:VDTNGCJR","canonical_record":{"source":{"id":"1107.4890","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-25T10:55:58Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"b3637bbeb398940ee6c186f59251e886cdd60cee725cf7ece8464ca466e4c132","abstract_canon_sha256":"172f9085c12d9380153e75f4fd1181b21262721e38cbeba87ae87439b331ca9b"},"schema_version":"1.0"},"canonical_sha256":"a8e6d30931f11aaccfd4c27fcf8c8fd3f4479bc5b03ece0fee7b4b5ac6638256","source":{"kind":"arxiv","id":"1107.4890","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.4890","created_at":"2026-05-18T04:16:54Z"},{"alias_kind":"arxiv_version","alias_value":"1107.4890v1","created_at":"2026-05-18T04:16:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.4890","created_at":"2026-05-18T04:16:54Z"},{"alias_kind":"pith_short_12","alias_value":"VDTNGCJR6ENK","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"VDTNGCJR6ENKZT6U","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"VDTNGCJR","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:VDTNGCJR6ENKZT6UYJ747DEP2P","target":"record","payload":{"canonical_record":{"source":{"id":"1107.4890","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-25T10:55:58Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"b3637bbeb398940ee6c186f59251e886cdd60cee725cf7ece8464ca466e4c132","abstract_canon_sha256":"172f9085c12d9380153e75f4fd1181b21262721e38cbeba87ae87439b331ca9b"},"schema_version":"1.0"},"canonical_sha256":"a8e6d30931f11aaccfd4c27fcf8c8fd3f4479bc5b03ece0fee7b4b5ac6638256","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:16:54.354141Z","signature_b64":"ow0ToNjpwrPC2VAIj7pePxQprQIcqL7/alBHFlU3veidIGNIibB0R1CmGp9uoZshpsC8nJ2PqQihJYXcbRbWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8e6d30931f11aaccfd4c27fcf8c8fd3f4479bc5b03ece0fee7b4b5ac6638256","last_reissued_at":"2026-05-18T04:16:54.353486Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:16:54.353486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.4890","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:16:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KOIz+bx2ASCtQuuTLmZGXld3lc52x9cw0Oyzx/5WWup7nBj58lT1kBqXJrhBA1FhkS+JlHSJpCaIPrTMws8XAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T21:05:52.114546Z"},"content_sha256":"56e0030b72c013274a7d80acc4704da5ddb9fee1e4842c7f51ac274115442e2e","schema_version":"1.0","event_id":"sha256:56e0030b72c013274a7d80acc4704da5ddb9fee1e4842c7f51ac274115442e2e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:VDTNGCJR6ENKZT6UYJ747DEP2P","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counting Square-Free Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.NT","authors_text":"Jakub Pawlewicz","submitted_at":"2011-07-25T10:55:58Z","abstract_excerpt":"The main topic of this contribution is the problem of counting square-free numbers not exceeding $n$. Before this work we were able to do it in time (Comparing to the Big-O notation, Soft-O ($\\softO$) ignores logarithmic factors) $\\softO(\\sqrt{n})$. Here, the algorithm with time complexity $\\softO(n^{2/5})$ and with memory complexity $\\softO(n^{1/5})$ is presented. Additionally, a parallel version is shown, which achieves full scalability.\n  As of now the highest computed value was for $n=10^{17}$. Using our implementation we were able to calculate the value for $n=10^{36}$ on a cluster."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.4890","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:16:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dBwkPa1I3LT7OS5168xSjE20pMFs9pjmIqLelvpvAUMKqex3EoS2lfLts01vNYLPaqr7fAYlQ1tQmU5AO19SAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T21:05:52.114987Z"},"content_sha256":"81078cca95c156592054dcacbe9ba19649693361038893c0f0294780ce178e10","schema_version":"1.0","event_id":"sha256:81078cca95c156592054dcacbe9ba19649693361038893c0f0294780ce178e10"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VDTNGCJR6ENKZT6UYJ747DEP2P/bundle.json","state_url":"https://pith.science/pith/VDTNGCJR6ENKZT6UYJ747DEP2P/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VDTNGCJR6ENKZT6UYJ747DEP2P/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T21:05:52Z","links":{"resolver":"https://pith.science/pith/VDTNGCJR6ENKZT6UYJ747DEP2P","bundle":"https://pith.science/pith/VDTNGCJR6ENKZT6UYJ747DEP2P/bundle.json","state":"https://pith.science/pith/VDTNGCJR6ENKZT6UYJ747DEP2P/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VDTNGCJR6ENKZT6UYJ747DEP2P/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VDTNGCJR6ENKZT6UYJ747DEP2P","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"172f9085c12d9380153e75f4fd1181b21262721e38cbeba87ae87439b331ca9b","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-25T10:55:58Z","title_canon_sha256":"b3637bbeb398940ee6c186f59251e886cdd60cee725cf7ece8464ca466e4c132"},"schema_version":"1.0","source":{"id":"1107.4890","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.4890","created_at":"2026-05-18T04:16:54Z"},{"alias_kind":"arxiv_version","alias_value":"1107.4890v1","created_at":"2026-05-18T04:16:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.4890","created_at":"2026-05-18T04:16:54Z"},{"alias_kind":"pith_short_12","alias_value":"VDTNGCJR6ENK","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"VDTNGCJR6ENKZT6U","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"VDTNGCJR","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:81078cca95c156592054dcacbe9ba19649693361038893c0f0294780ce178e10","target":"graph","created_at":"2026-05-18T04:16:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The main topic of this contribution is the problem of counting square-free numbers not exceeding $n$. Before this work we were able to do it in time (Comparing to the Big-O notation, Soft-O ($\\softO$) ignores logarithmic factors) $\\softO(\\sqrt{n})$. Here, the algorithm with time complexity $\\softO(n^{2/5})$ and with memory complexity $\\softO(n^{1/5})$ is presented. Additionally, a parallel version is shown, which achieves full scalability.\n  As of now the highest computed value was for $n=10^{17}$. Using our implementation we were able to calculate the value for $n=10^{36}$ on a cluster.","authors_text":"Jakub Pawlewicz","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-25T10:55:58Z","title":"Counting Square-Free Numbers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.4890","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:56e0030b72c013274a7d80acc4704da5ddb9fee1e4842c7f51ac274115442e2e","target":"record","created_at":"2026-05-18T04:16:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"172f9085c12d9380153e75f4fd1181b21262721e38cbeba87ae87439b331ca9b","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-07-25T10:55:58Z","title_canon_sha256":"b3637bbeb398940ee6c186f59251e886cdd60cee725cf7ece8464ca466e4c132"},"schema_version":"1.0","source":{"id":"1107.4890","kind":"arxiv","version":1}},"canonical_sha256":"a8e6d30931f11aaccfd4c27fcf8c8fd3f4479bc5b03ece0fee7b4b5ac6638256","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a8e6d30931f11aaccfd4c27fcf8c8fd3f4479bc5b03ece0fee7b4b5ac6638256","first_computed_at":"2026-05-18T04:16:54.353486Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:16:54.353486Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ow0ToNjpwrPC2VAIj7pePxQprQIcqL7/alBHFlU3veidIGNIibB0R1CmGp9uoZshpsC8nJ2PqQihJYXcbRbWDg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:16:54.354141Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.4890","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:56e0030b72c013274a7d80acc4704da5ddb9fee1e4842c7f51ac274115442e2e","sha256:81078cca95c156592054dcacbe9ba19649693361038893c0f0294780ce178e10"],"state_sha256":"f1cd1d9fd1c3803e38cbd5e02d61aba1e0da35295f7444f6caa2a8490f214db0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t6/icgMVtexJyJobhOpJTy50s5J3QBzzd68pTnlcOK3PSEbwjZ9AdM5OEvtM2Up8TI6YLReg/FimSjYn4khiDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T21:05:52.118005Z","bundle_sha256":"3b20635e8358865aee44d5dbdc4297b4a94e1ba8cef09c049f2d1a1d519ef161"}}