{"id":"W1632722589","doi":"10.1142/s1793042115500761","title":"Amicable pairs and aliquot cycles on average","year":2015,"lang":"en","type":"article","venue":"International Journal of Number Theory","topic":"Algebraic Geometry and Number Theory","field":"Mathematics","cited_by":6,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Lethbridge","funders":"","keywords":"Function (biology); Elliptic curve; Order (exchange); Upper and lower bounds; Generating function","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":["insufficient_payload"],"consensus_categories":[],"category_scores_codex":[0.001744189,0.0001374472,0.0002211881,0.0001335116,0.00003422732,0.00005495932,0.0003837035,0.00007049047,0.001680608],"category_scores_gemma":[0.001458112,0.0001098082,0.0001077557,0.00006413913,0.0001043668,0.0002424383,0.0000850102,0.0002586284,0.0001263719],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00008417118,"about_ca_system_score_gemma":0.00005877766,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000002397401,"about_ca_topic_score_gemma":5.998787e-7,"domain_scores_codex":[0.998522,0.0002342096,0.0004038523,0.0001164462,0.0005806733,0.000142844],"domain_scores_gemma":[0.9976975,0.001284627,0.0003483171,0.0001371318,0.000366391,0.0001660056],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.001163082,0.0003608952,0.002070763,0.00001342817,0.0005057506,0.0003026314,0.00163019,0.00003311944,0.00009508146,0.9491666,0.03756692,0.007091534],"study_design_scores_gemma":[0.0009686229,0.00009795105,0.0004068358,0.0001212108,0.00002736242,0.0009678216,0.0007850445,0.00001685132,0.0008954444,0.9834909,0.01209468,0.0001272072],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9270368,0.00009182942,0.003865231,0.0009361092,0.001281062,0.00006585583,0.00003781974,0.00003352091,0.06665172],"genre_scores_gemma":[0.9938393,0.00002286424,0.002267776,0.0005686619,0.0005364853,0.000001732795,0.000003637861,0.00002244246,0.002737076],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.06680247,"threshold_uncertainty_score":0.999232,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.04224231027477038,"score_gpt":0.3267979984995437,"score_spread":0.2845556882247733,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}