We extend to Segal–Piatetski-Shapiro sequences previous results on the Luca–Schinzel question. Namely, we prove that for any real c larger than 1, sequence $$(\sum _{m\le n} \varphi (\lfloor m^c \rfloor ) /\lfloor )_{n\ge 1}$$ is dense modulo where $$\varphi $$ denotes Euler’s totient function. The main part of proof consists in showing when R a large integer, residues $$\lfloor contains block ...

We prove that if f(x) = ∑n−1 k=0 akx k is a polynomial with no cyclotomic factors whose coefficients satisfy ak ≡ 1 mod 2 for 0 ≤ k < n, then Mahler’s measure of f satisfies log M(f) ≥ log 5 4 ( 1 − 1 n ) . This resolves a problem of D. H. Lehmer [12] for the class of polynomials with odd coefficients. We also prove that if f has odd coefficients, degree n−1, and at least one noncyclotomic fact...

Distributions of the values of some arithmetical functions by P. ERDÖS (Budapest) and A. SCHINZEL (Warszawa) § 1. Y. Wang and A. Schinzel proved. by Brun's method, the following theorem ([3]) : For any given sequence of h nova-negative-numbers a,, a 2 ,. . ., a h and > 0, there exist positive constants e = c(a, E) and x o = x0 (a, E) such that the number of positive integers n, <. x satisfying ...

In the theory of generalized Davenport{Schinzel sequences one estimates the maximum lengths of nite sequences containing no subsequence of a given pattern. Here we investigate a further generalization, in which the class of sequences is extended to the class of colored trees. We determine exactly the extremal functions associated with the properly 2-colored path of four vertices and with the mo...