{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:YQPFWJJRVVQDLD7PWU5KDKW2PI","short_pith_number":"pith:YQPFWJJR","canonical_record":{"source":{"id":"1702.02616","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-02-08T21:29:42Z","cross_cats_sorted":[],"title_canon_sha256":"09eef496a2d09d7e131e2c247ccd31daeacb547b9eabda5b460edd944500bca5","abstract_canon_sha256":"1b84eeb926f4ad4399a965252976c1a7e1c76bc80cd06f3ed742a605423e8983"},"schema_version":"1.0"},"canonical_sha256":"c41e5b2531ad60358fefb53aa1aada7a04e5d424b7645d4e38b10d5c1474bbc1","source":{"kind":"arxiv","id":"1702.02616","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02616","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02616v1","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02616","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"pith_short_12","alias_value":"YQPFWJJRVVQD","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YQPFWJJRVVQDLD7P","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YQPFWJJR","created_at":"2026-05-18T12:31:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:YQPFWJJRVVQDLD7PWU5KDKW2PI","target":"record","payload":{"canonical_record":{"source":{"id":"1702.02616","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-02-08T21:29:42Z","cross_cats_sorted":[],"title_canon_sha256":"09eef496a2d09d7e131e2c247ccd31daeacb547b9eabda5b460edd944500bca5","abstract_canon_sha256":"1b84eeb926f4ad4399a965252976c1a7e1c76bc80cd06f3ed742a605423e8983"},"schema_version":"1.0"},"canonical_sha256":"c41e5b2531ad60358fefb53aa1aada7a04e5d424b7645d4e38b10d5c1474bbc1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:03.860073Z","signature_b64":"hKP8hERBHGZef8K8JCYoCGDxfGY354QEZz2SfG1XmmcFAM0CJN20YnA1yS1oDfc/HTGhpWH7NPwE1H5Vif/mBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c41e5b2531ad60358fefb53aa1aada7a04e5d424b7645d4e38b10d5c1474bbc1","last_reissued_at":"2026-05-18T00:51:03.859721Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:03.859721Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1702.02616","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":"2026-05-18T00:51:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZX3J9vXXYAMQwrcVfNh+HrgPyTj60meCLwnIn89J+Q+cZndR/Cast48w0ryiyL7nDyBRA66xOeQc7Zz3G+1TCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:19:31.154398Z"},"content_sha256":"fee53cd8c1b88c7a67b0f7f413473fe9769fbd3108d1ee3665be3560af53d173","schema_version":"1.0","event_id":"sha256:fee53cd8c1b88c7a67b0f7f413473fe9769fbd3108d1ee3665be3560af53d173"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:YQPFWJJRVVQDLD7PWU5KDKW2PI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Enumeration of groups whose order factorises in at most 4 primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Bettina Eick","submitted_at":"2017-02-08T21:29:42Z","abstract_excerpt":"Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02616","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":"2026-05-18T00:51:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mmAFwU1g2sQAO+7bG/g+VbemvcE0A/AuLRq8hTkvE+aEh7XLT8tOOD6nxZ6xY38KHAlrNARsi+yPFd5flY3JBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T18:19:31.155171Z"},"content_sha256":"da82942c092b683e4439b09d13aa58b23f3042b778a6d9c407468be9229edc93","schema_version":"1.0","event_id":"sha256:da82942c092b683e4439b09d13aa58b23f3042b778a6d9c407468be9229edc93"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YQPFWJJRVVQDLD7PWU5KDKW2PI/bundle.json","state_url":"https://pith.science/pith/YQPFWJJRVVQDLD7PWU5KDKW2PI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YQPFWJJRVVQDLD7PWU5KDKW2PI/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":"2026-05-31T18:19:31Z","links":{"resolver":"https://pith.science/pith/YQPFWJJRVVQDLD7PWU5KDKW2PI","bundle":"https://pith.science/pith/YQPFWJJRVVQDLD7PWU5KDKW2PI/bundle.json","state":"https://pith.science/pith/YQPFWJJRVVQDLD7PWU5KDKW2PI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YQPFWJJRVVQDLD7PWU5KDKW2PI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:YQPFWJJRVVQDLD7PWU5KDKW2PI","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":"1b84eeb926f4ad4399a965252976c1a7e1c76bc80cd06f3ed742a605423e8983","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-02-08T21:29:42Z","title_canon_sha256":"09eef496a2d09d7e131e2c247ccd31daeacb547b9eabda5b460edd944500bca5"},"schema_version":"1.0","source":{"id":"1702.02616","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02616","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02616v1","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02616","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"pith_short_12","alias_value":"YQPFWJJRVVQD","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YQPFWJJRVVQDLD7P","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YQPFWJJR","created_at":"2026-05-18T12:31:56Z"}],"graph_snapshots":[{"event_id":"sha256:da82942c092b683e4439b09d13aa58b23f3042b778a6d9c407468be9229edc93","target":"graph","created_at":"2026-05-18T00:51:03Z","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":"Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$.","authors_text":"Bettina Eick","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-02-08T21:29:42Z","title":"Enumeration of groups whose order factorises in at most 4 primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02616","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:fee53cd8c1b88c7a67b0f7f413473fe9769fbd3108d1ee3665be3560af53d173","target":"record","created_at":"2026-05-18T00:51:03Z","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":"1b84eeb926f4ad4399a965252976c1a7e1c76bc80cd06f3ed742a605423e8983","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-02-08T21:29:42Z","title_canon_sha256":"09eef496a2d09d7e131e2c247ccd31daeacb547b9eabda5b460edd944500bca5"},"schema_version":"1.0","source":{"id":"1702.02616","kind":"arxiv","version":1}},"canonical_sha256":"c41e5b2531ad60358fefb53aa1aada7a04e5d424b7645d4e38b10d5c1474bbc1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c41e5b2531ad60358fefb53aa1aada7a04e5d424b7645d4e38b10d5c1474bbc1","first_computed_at":"2026-05-18T00:51:03.859721Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:03.859721Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"hKP8hERBHGZef8K8JCYoCGDxfGY354QEZz2SfG1XmmcFAM0CJN20YnA1yS1oDfc/HTGhpWH7NPwE1H5Vif/mBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:03.860073Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.02616","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fee53cd8c1b88c7a67b0f7f413473fe9769fbd3108d1ee3665be3560af53d173","sha256:da82942c092b683e4439b09d13aa58b23f3042b778a6d9c407468be9229edc93"],"state_sha256":"bf170fe088d20df0c57471d4ef8ca5b18121a82bac493e9c32db9198bed69ed0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mRJ3OkVXE8H2QhZA3bzrIobHU8aaOEG9bCz+iJ+pu+xDmUPLz681URBL3EEDYyo2mL/SI6Pc0C/jo+LkHMn7Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T18:19:31.159770Z","bundle_sha256":"66210ca0e2a5bfa051da34e452f4c8ebd5c573847d616e6fb81769b076fd9cff"}}