{"id":"W7108465371","doi":"10.61091/jcmcc128-22","title":"On the numerical semigroup generated by \\(\\left\\{b^{k+n+i}+\\frac{b^{n+i}-1}{b-1}\\mid i\\in\\mathbb{N}\\right\\}\\)","year":2025,"lang":"","type":"article","venue":"Journal of Combinatorial Mathematics and Combinatorial Computing","topic":"Commutative Algebra and Its Applications","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"Basic and Applied Basic Research Foundation of Guangdong Province","keywords":"Semigroup; Embedding; Dimension (graph theory); Numerical semigroup; Algebra over a field","routes":{"ca_aff":false,"ca_fund":false,"ca_venue":true,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":["metaepi_narrow","research_integrity"],"consensus_categories":[],"category_scores_codex":[0.00489102,0.001113552,0.002482326,0.0006494821,0.001190923,0.0008812489,0.001812015,0.000668221,0.0001708922],"category_scores_gemma":[0.003486292,0.0008459325,0.0006238632,0.001749313,0.0004331895,0.0003221875,0.0007871968,0.002935383,0.00002138927],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0005403494,"about_ca_system_score_gemma":0.0006620052,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0000200083,"about_ca_topic_score_gemma":0.000001619937,"domain_scores_codex":[0.9915107,0.0009921048,0.004227705,0.0007104864,0.001506145,0.001052851],"domain_scores_gemma":[0.9820138,0.01220166,0.003048442,0.001105798,0.001247279,0.0003829733],"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":[0.0002756166,0.003691118,0.0001004499,0.0004588368,0.0005619029,0.00003339532,0.002441243,0.00004371664,0.001263157,0.9732612,0.01688923,0.0009800831],"study_design_scores_gemma":[0.007158887,0.001186248,0.00006248019,0.002419878,0.0005171011,0.00007762934,0.001154649,0.01690528,0.00423217,0.9607012,0.004804737,0.0007797831],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9402204,0.00304077,0.02042667,0.005164017,0.02061065,0.002105011,0.00003616036,0.00008351028,0.008312873],"genre_scores_gemma":[0.996465,0.0003032835,0.001566029,0.0003519707,0.001004097,0.00001579966,0.000005380545,0.0001172016,0.0001712622],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.05624464,"threshold_uncertainty_score":0.9993991,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01391938853613487,"score_gpt":0.2814264520745566,"score_spread":0.2675070635384217,"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."}}