Further Results on a Curious Arithmetic Function
Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for...
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| Format: | Article |
| Language: | English |
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Wiley
2020-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2020/1894162 |
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| author | Long Chen Kaimin Cheng Tingting Wang |
| author_facet | Long Chen Kaimin Cheng Tingting Wang |
| author_sort | Long Chen |
| collection | DOAJ |
| description | Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime. |
| format | Article |
| id | doaj-art-0012cf49e9e9435ab917c793c8b0b521 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-0012cf49e9e9435ab917c793c8b0b5212025-02-03T01:05:21ZengWileyJournal of Mathematics2314-46292314-47852020-01-01202010.1155/2020/18941621894162Further Results on a Curious Arithmetic FunctionLong Chen0Kaimin Cheng1Tingting Wang2School of Mathematics and Computer Science, Panzhihua University, Panzhihua 617000, ChinaSchool of Mathematics and Information, China West Normal University, Nanchong 637002, ChinaChengdu Experimental Foreign Language School, Chengdu 610061, ChinaLet p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.http://dx.doi.org/10.1155/2020/1894162 |
| spellingShingle | Long Chen Kaimin Cheng Tingting Wang Further Results on a Curious Arithmetic Function Journal of Mathematics |
| title | Further Results on a Curious Arithmetic Function |
| title_full | Further Results on a Curious Arithmetic Function |
| title_fullStr | Further Results on a Curious Arithmetic Function |
| title_full_unstemmed | Further Results on a Curious Arithmetic Function |
| title_short | Further Results on a Curious Arithmetic Function |
| title_sort | further results on a curious arithmetic function |
| url | http://dx.doi.org/10.1155/2020/1894162 |
| work_keys_str_mv | AT longchen furtherresultsonacuriousarithmeticfunction AT kaimincheng furtherresultsonacuriousarithmeticfunction AT tingtingwang furtherresultsonacuriousarithmeticfunction |