Construction of normal numbers using the distribution of the \(kth\) largest prime factor

Authors

  • J.-M. De koninck
  • I. Katai

Keywords:

normal numbers, largest prime factor

Abstract

Given an integer q>1, a q-normal number is an irrational number r such that any preassigned sequence of k digits occurs in the q-ary expansion of r at the expected frequency, namely 1/q^k. In a recent paper, we constructed a large family of normal numbers, showing in particular that the numbers 0.P(2)P(3)P(4)... and 0.P(2+1)P(3+1)P(5+1)...P(p+1)..., where P(n) stands for the largest prime factor of n, are normal numbers, thereby answering in the affirmative a question raised by Igor Shparlinski. Here, we show that, given any fixed integer k>1, the numbers 0.P_k(2)P_k(3)P_k(4)... and 0.P_k(2+1)P_k(3+1)P_k(5+1)...P_k(p+1)..., where P_k(n) stands for the k-th largest prime factor of n, are normal numbers. Much more general results are also obtained. 10.1017/S0004972712000998

Published

2013-07-23

Issue

Vol. 88 No. 1 (2013)

Section

Articles
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