{"id":"W2117095608","doi":"10.1142/s1793042108001614","title":"BERNDT'S CURIOUS FORMULA","year":2008,"lang":"en","type":"article","venue":"International Journal of Number Theory","topic":"Advanced Mathematical Identities","field":"Mathematics","cited_by":1,"is_retracted":false,"has_abstract":true,"ca_institutions":"Carleton University","funders":"","keywords":"Mathematics; Ramanujan's sum; Integer (computer science); Pure mathematics; Arithmetic function; Arithmetic; Combinatorics; Discrete mathematics","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":[0.000419564,0.0001219281,0.0002425923,0.0001026683,0.00005378175,0.00002927468,0.0005542785,0.00005082596,0.00412123],"category_scores_gemma":[0.001719324,0.0000990866,0.0002116519,0.00005132056,0.0001388758,0.0004726956,0.00009275698,0.0002231615,0.0001997707],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0001069668,"about_ca_system_score_gemma":0.00006070705,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000001170141,"about_ca_topic_score_gemma":0.000001135133,"domain_scores_codex":[0.9984133,0.00006345163,0.0005800435,0.00008192356,0.0007128922,0.0001484248],"domain_scores_gemma":[0.9974881,0.00123697,0.0004767672,0.000143107,0.000571663,0.00008333497],"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":[0.0001451371,0.000310658,0.0003414208,0.00002114326,0.0003540989,0.0008061237,0.001478717,0.000008375788,0.0001687524,0.9581398,0.03702338,0.00120242],"study_design_scores_gemma":[0.0004868372,0.00002427661,0.0001313239,0.0001040006,0.00002531128,0.004509873,0.0001764401,0.00001177378,0.0007048369,0.9837207,0.0100013,0.0001033201],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.6661847,0.0002674677,0.1474623,0.001204651,0.003503885,0.0002273628,0.00009906408,0.0001380639,0.1809124],"genre_scores_gemma":[0.9327036,0.0001289883,0.04645327,0.0003044328,0.001015665,0.000005341655,0.00000374298,0.000051054,0.01933389],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.2665189,"threshold_uncertainty_score":0.9967892,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.05076704654807616,"score_gpt":0.3694805102014368,"score_spread":0.3187134636533606,"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."}}