On a thin set of integers involving the largest prime factor function
For each integer n≥2, let P(n) denote its largest prime factor. Let S:={n≥2:n does not divide P(n)!} and S(x):=#{n≤x:n∈S}. Erdős (1991) conjectured that S is a set of zero density. This was proved by Kastanas (1994) who established that S(x)=O(x/logx). Recently, Akbik (1999) proved that S(x)=O(x exp...
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| Format: | Article |
| Language: | English |
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Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S016117120320418X |
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| _version_ | 1832552346841776128 |
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| author | Jean-Marie De Koninck Nicolas Doyon |
| author_facet | Jean-Marie De Koninck Nicolas Doyon |
| author_sort | Jean-Marie De Koninck |
| collection | DOAJ |
| description | For each integer n≥2, let P(n) denote its largest prime factor. Let S:={n≥2:n does not divide P(n)!} and S(x):=#{n≤x:n∈S}. Erdős (1991) conjectured that S is a set of zero density. This was proved by Kastanas (1994) who established that S(x)=O(x/logx). Recently, Akbik (1999) proved that S(x)=O(x exp{−(1/4)logx}). In this paper, we show that S(x)=x exp{−(2+o(1))×log x log log x}. We also investigate small and large gaps among the elements of S and state some conjectures. |
| format | Article |
| id | doaj-art-3345b68e67e94663bcc9f02610ec81e0 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2003-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-3345b68e67e94663bcc9f02610ec81e02025-02-03T05:58:47ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252003-01-012003191185119210.1155/S016117120320418XOn a thin set of integers involving the largest prime factor functionJean-Marie De Koninck0Nicolas Doyon1Département de Mathématiques et de Statistique, Université Laval, Québec, Québec G1K 7P4, CanadaDépartement de Mathématiques et de Statistique, Université de Montréal, Québec, Montréal H3C 3J7, CanadaFor each integer n≥2, let P(n) denote its largest prime factor. Let S:={n≥2:n does not divide P(n)!} and S(x):=#{n≤x:n∈S}. Erdős (1991) conjectured that S is a set of zero density. This was proved by Kastanas (1994) who established that S(x)=O(x/logx). Recently, Akbik (1999) proved that S(x)=O(x exp{−(1/4)logx}). In this paper, we show that S(x)=x exp{−(2+o(1))×log x log log x}. We also investigate small and large gaps among the elements of S and state some conjectures.http://dx.doi.org/10.1155/S016117120320418X |
| spellingShingle | Jean-Marie De Koninck Nicolas Doyon On a thin set of integers involving the largest prime factor function International Journal of Mathematics and Mathematical Sciences |
| title | On a thin set of integers involving the largest prime factor function |
| title_full | On a thin set of integers involving the largest prime factor function |
| title_fullStr | On a thin set of integers involving the largest prime factor function |
| title_full_unstemmed | On a thin set of integers involving the largest prime factor function |
| title_short | On a thin set of integers involving the largest prime factor function |
| title_sort | on a thin set of integers involving the largest prime factor function |
| url | http://dx.doi.org/10.1155/S016117120320418X |
| work_keys_str_mv | AT jeanmariedekoninck onathinsetofintegersinvolvingthelargestprimefactorfunction AT nicolasdoyon onathinsetofintegersinvolvingthelargestprimefactorfunction |