Arithmetic properties of positive integers with fixed digit sum (Q879625)

The author proves 7 propositions about the set \(A_{b,s}\) of all positive integers \(n\) which are not multiples of \(b\) and whose sum of digits in base \(b\) is precisely \(s\). For instance: Proposition 2: Let \(P(n)\) be the largest prime factor of \(n\). Then for every \(\varepsilon > 0\) there exist infinitely many positive integers \(n\in A_{b,s}\) with \(P(n)< n^{\varepsilon}\); Propositions 4 and 5: There exist infinitely many positive integers \(n\in A_{b,s}\) for which \(\omega (n)>\exp(\frac{c_{5}\log_{2}n}{\log_{3}n})\) and \(\Omega(n)=o(\log n)\) where \(\omega (n) \) means as usually the number of distinct prime factors of \(n\) ; \(\Omega(n)\) - the same, but counting the multiplicity of the primes, \(c_{5}\) is a positive computable constant depending only on \(b\) and \(s\). The author proposes the following problem: Prove or disprove that \(\lim_{{n\rightarrow\infty} \atop{n\in A_{b,s}}} \frac{\omega(n)\log_{2}n}{\log n}=0\).