{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,24]],"date-time":"2025-09-24T00:09:25Z","timestamp":1758672565993,"version":"3.44.0"},"reference-count":18,"publisher":"World Scientific Pub Co Pte Ltd","issue":"07","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12071311"],"award-info":[{"award-number":["12071311"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:p> Let [Formula: see text] be the numerical semigroup generated by relatively prime positive integers [Formula: see text]. The quotient of [Formula: see text] with respect to a positive integer p is defined by [Formula: see text]. The quotient [Formula: see text] is known to be a semigroup, but is hard to study. When p is a positive divisor of [Formula: see text], we reduce the computation of the Ap\u00e9ry set of [Formula: see text] in [Formula: see text] to a simple minimization problem. This allows us to obtain closed formulas of the Frobenius number of the quotient for some special numerical semigroups. These include the cases when [Formula: see text] is the almost arithmetic progressions, the almost arithmetic progressions with initial gaps. In particular, we partially solve an open problem proposed by Adeniran et\u00a0al. [On the genus of a quotient of a numerical semigroup, Semigroup Forum.\u00a098 (2019) 690\u2013700]. <\/jats:p>","DOI":"10.1142\/s0218196725500341","type":"journal-article","created":{"date-parts":[[2025,7,25]],"date-time":"2025-07-25T06:25:02Z","timestamp":1753424702000},"page":"1079-1090","source":"Crossref","is-referenced-by-count":0,"title":["On quotients of numerical semigroups for almost arithmetic progressions"],"prefix":"10.1142","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9609-1880","authenticated-orcid":false,"given":"Feihu","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.\u00a0R.\u00a0China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2025,8,19]]},"reference":[{"key":"S0218196725500341BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/s00233-018-9989-3"},{"key":"S0218196725500341BIB002","doi-asserted-by":"publisher","DOI":"10.2307\/2371684"},{"key":"S0218196725500341BIB003","first-page":"215","volume":"211","author":"Brauer A.","year":"1962","journal-title":"J. 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