De Koninck, Jean-Marie; Kátai, Imre Construction of normal numbers using the distribution of the \(k\)th largest prime factor. (English) Zbl 1282.11089 Bull. Aust. Math. Soc. 88, No. 1, 158-168 (2013). Summary: Given an integer \(q\geq 2\), a \(q\)-normal number is an irrational number \(\eta \) such that any preassigned sequence of \(\ell \) digits occurs in the \(q\)-ary expansion of \(\eta \) at the expected frequency, namely \(1/q^\ell \). In a recent paper we constructed a large family of normal numbers, showing in particular that, if \(P(n)\) stands for the largest prime factor of \(n\), then the number \(0.P(2)P(3)P(4)\ldots ,\) the concatenation of the numbers \(P(2), P(3), P(4), \ldots ,\) each represented in base \(q\), is a \(q\)-normal number, thereby answering in the affirmative a question raised by Igor Shparlinski. We also showed that \(0.P(2+1)P(3+1)P(5+1) \ldots P(p+1)\ldots ,\) where \(p\) runs through the sequence of primes, is a \(q\)-normal number. Here, we show that, given any fixed integer \(k\geq 2\), the numbers \(0.P_k(2)P_k(3)P_k(4)\ldots \) and \(0. P_k(2+1)P_k(3+1)P_k(5+1) \ldots P_k(p+1)\ldots ,\) where \(P_k(n)\) stands for the \(k\)th largest prime factor of \(n\), are \(q\)-normal numbers. These results are part of more general statements. Cited in 1 ReviewCited in 2 Documents MSC: 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11N37 Asymptotic results on arithmetic functions 11A41 Primes Keywords:normal numbers; largest prime factor PDFBibTeX XMLCite \textit{J.-M. De Koninck} and \textit{I. Kátai}, Bull. Aust. Math. Soc. 88, No. 1, 158--168 (2013; Zbl 1282.11089) Full Text: DOI References: [1] De Koninck, Analytic Number Theory: Exploring the Anatomy of Integers (2012) · Zbl 1247.11001 [2] Halberstam, Sieve Methods (1974) [3] DOI: 10.1007/BF02367950 [4] DOI: 10.1017/S0004972711002322 · Zbl 1231.11086 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.