Victor Manuel Aricheta, Jerome Dimabayao, Hazel Joy Shi
{"title":"(4k,k)奇异超分区奇偶性的密度结果","authors":"Victor Manuel Aricheta, Jerome Dimabayao, Hazel Joy Shi","doi":"10.1016/j.jnt.2024.03.016","DOIUrl":null,"url":null,"abstract":"<div><p>The <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>i</mi><mo>)</mo></math></span>-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to 6<em>k</em>. In this paper we consider the parity of the number <span><math><msub><mrow><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>i</mi><mo>)</mo></math></span>-singular overpartitions of <em>n</em>. In particular, we give a sufficient condition on even values of <em>k</em> so that the values of <span><math><msub><mrow><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>4</mn><mi>k</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are almost always even. Furthermore, we show that for odd values of <span><math><mi>k</mi><mo>≤</mo><mn>23</mn></math></span>, <span><math><mi>k</mi><mo>≠</mo><mn>19</mn></math></span>, certain subsequences of <span><math><msub><mrow><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>4</mn><mi>k</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are almost always even.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Density results for the parity of (4k,k)-singular overpartitions\",\"authors\":\"Victor Manuel Aricheta, Jerome Dimabayao, Hazel Joy Shi\",\"doi\":\"10.1016/j.jnt.2024.03.016\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>i</mi><mo>)</mo></math></span>-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to 6<em>k</em>. In this paper we consider the parity of the number <span><math><msub><mrow><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>k</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>i</mi><mo>)</mo></math></span>-singular overpartitions of <em>n</em>. In particular, we give a sufficient condition on even values of <em>k</em> so that the values of <span><math><msub><mrow><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>4</mn><mi>k</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are almost always even. Furthermore, we show that for odd values of <span><math><mi>k</mi><mo>≤</mo><mn>23</mn></math></span>, <span><math><mi>k</mi><mo>≠</mo><mn>19</mn></math></span>, certain subsequences of <span><math><msub><mrow><mover><mrow><mi>C</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>4</mn><mi>k</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are almost always even.</p></div>\",\"PeriodicalId\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-04-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24000878\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24000878","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Density results for the parity of (4k,k)-singular overpartitions
The -singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to 6k. In this paper we consider the parity of the number of -singular overpartitions of n. In particular, we give a sufficient condition on even values of k so that the values of are almost always even. Furthermore, we show that for odd values of , , certain subsequences of are almost always even.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access.
JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions.
Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.