Normal numbers generated using the smallest prime factor function

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• 作者：Jean-Marie De Koninck (1)
  Imre Kátai (2)
  
• 关键词：Normal numbers ; Smallest prime factor ; 11K16 ; 11N37 ; 11N41
• 刊名：Annales math篓娄matiques du Qu篓娄bec
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：38
• 期：2
• 页码：133-144
• 全文大小：171 KB
• 参考文献：1. Bassily, N.L., Kátai, I.: Distribution of consecutive digits in the \(q\) -ary expansions of some sequences of integers. J. Math. Sci. 78(1), 11-7 (1996) CrossRef
  2. Davenport, H., Erd?s, P.: Note on normal decimals. Can. J. Math. 4, 58-3 (1952) CrossRef
  3. De Koninck, J.M., Kátai, I.: On a problem on normal numbers raised by Igor Shparlinski. Bull. Aust. Math. Soc. 84, 337-49 (2011) CrossRef
  4. De Koninck, J.M., Kátai, I.: Some new methods for constructing normal numbers. Ann. Sci. Math. Québec 36(2), 349-59 (2012)
  5. De Koninck, J.M., Kátai, I.: Construction of normal numbers using the distribution of the \(k\) -th largest prime factor. J. Aust. Math. Soc. 88, 158-68 (2013) CrossRef
  6. Elliott, P.D.T.A.: Probabilistic Number Theory I, Mean Value Theorems. Springer, Berlin (1979)
  7. Halberstam, H.H., Richert, H.E.: Sieve Methods. Academic Press, London (1974)
  
• 作者单位：Jean-Marie De Koninck (1)
  Imre Kátai (2)
  
  1. Dép. de mathématiques et de statistique, Université Laval, Québec, G1V 0A6, Canada
  2. Computer Algebra Department, E?tv?s Loránd University, Pázmány Péter Sétány I/C, Budapest, 1117, Hungary
  
• ISSN：2195-4763

文摘

In a series of papers, we constructed large families of normal numbers using the distribution of the values of the largest prime factor function. Here, letting \(p(n)\) stand for the smallest prime factor of \(n\) , we show how a concatenation of the successive values of \(p(n)\) can yield a normal number in any given basis \(q\ge 2\) . We further expand on this idea to create various large families of normal numbers.