{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:58:44Z","timestamp":1760234324375,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,3]],"date-time":"2021-05-03T00:00:00Z","timestamp":1620000000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let S and T be two numerical semigroups. We say that T is an I(S)-semigroup if T\u2216{0} is an ideal of S. Given k a positive integer, we denote by \u0394(k) the symmetric numerical semigroup generated by {2,2k+1}. In this paper we present a formula which calculates the number of I(S)-semigroups with genus g(\u0394(k))+h for some nonnegative integer h and which we will denote by i(\u0394(k),h). As a consequence, we obtain that the sequence {i(\u0394(k),h)}h\u2208N is never decreasing. Besides, it becomes stationary from a certain term.<\/jats:p>","DOI":"10.3390\/sym13050794","type":"journal-article","created":{"date-parts":[[2021,5,5]],"date-time":"2021-05-05T22:51:42Z","timestamp":1620255102000},"page":"794","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Counting the Ideals with a Given Genus of a Numerical Semigroup with Multiplicity Two"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0593-9434","authenticated-orcid":false,"given":"M. A.","family":"Moreno-Fr\u00edas","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Ciencias, Universidad de C\u00e1diz, Puerto Real, E-11510 C\u00e1diz, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 Carlos","family":"Rosales","sequence":"additional","affiliation":[{"name":"Departamento de \u00c1lgebra, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,3]]},"reference":[{"doi-asserted-by":"crossref","unstructured":"Rosales, J.C., and Garc\u00eda-S\u00e1nchez, P.A. (2009). Numerical Semigroups. 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Simon Stevin"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s00233-007-9014-8","article-title":"Fibonacci-like behavior of the number of numerical semigroups of a given genus","volume":"76","year":"2008","journal-title":"Semigroup Forum"},{"key":"ref_9","first-page":"255","article-title":"The set of numerical semigroups of a given genus","volume":"85","author":"Blanco","year":"2012","journal-title":"Forum Math."},{"key":"ref_10","first-page":"997","article-title":"Bounds on the number of numerical semigroups of a given genus","volume":"213","year":"2008","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1862","DOI":"10.1016\/j.jpaa.2009.12.031","article-title":"Improved bounds on the number of numerical semigroups of a given genus","volume":"214","author":"Elizalde","year":"2010","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1016","DOI":"10.1016\/j.jpaa.2011.10.038","article-title":"Counting numerical semigroups by genus and some cases a question of Wilf","volume":"216","author":"Kaplan","year":"2012","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1007\/s00233-009-9190-9","article-title":"Constructing numerical semigroup of a given genus","volume":"80","author":"Zhao","year":"2009","journal-title":"Semigroup Forum"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"634","DOI":"10.1007\/s00233-012-9456-5","article-title":"Fibonacci-like growth of numerical semigroups with a given genus","volume":"86","author":"Zhai","year":"2013","journal-title":"Semigroup Forum"},{"unstructured":"Moreno-Fr\u00edas, M.A., and Rosales, J.C. The ideals of a numerical semigroup. Counting the ideals with a given genus of an ordinary numerical semigroup Prepint.","key":"ref_15"},{"key":"ref_16","first-page":"21","article-title":"Mathematical question with their solutions","volume":"41","author":"Sylvester","year":"1884","journal-title":"Educational Times"},{"doi-asserted-by":"crossref","unstructured":"Ram\u00edrez Alfons\u00edn, J.L. (2005). The Diophantine Frobenius Problem, Oxford University Press.","key":"ref_17","DOI":"10.1093\/acprof:oso\/9780198568209.001.0001"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/794\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:56:51Z","timestamp":1760162211000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/794"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,3]]},"references-count":17,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["sym13050794"],"URL":"https:\/\/doi.org\/10.3390\/sym13050794","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,5,3]]}}}