{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:04:46Z","timestamp":1760231086241,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,23]],"date-time":"2022-08-23T00:00:00Z","timestamp":1661212800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11971476"],"award-info":[{"award-number":["11971476"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let a(n) be the number of non-isomorphic abelian groups of order n. In this paper, we study a symmetric form of the average value with respect to a(n) and prove an asymptotic formula. Furthermore, we study an analogue of the well-known Titchmarsh divisor problem involving a(n).<\/jats:p>","DOI":"10.3390\/sym14091755","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T02:55:34Z","timestamp":1661309734000},"page":"1755","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Symmetric Form of the Mean Value Involving Non-Isomorphic Abelian Groups"],"prefix":"10.3390","volume":"14","author":[{"given":"Haihong","family":"Fan","sequence":"first","affiliation":[{"name":"Department of Mathematics, China University of Mining and Technology, Beijing 100083, China"}]},{"given":"Wenguang","family":"Zhai","sequence":"additional","affiliation":[{"name":"Department of Mathematics, China University of Mining and Technology, Beijing 100083, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,23]]},"reference":[{"key":"ref_1","unstructured":"Ivic, A. (1985). 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