On a strange recursion of Golomb.
Let be the Ramanujan sum, i.e. , where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for (k = 1,2) are obtained. As an analogous problem, we evaluate (k = 1,2), where .
Given an integer , let be pairwise coprime integers , a family of nonempty proper subsets of with “enough” elements, and a function . Does there exist at least one prime such that divides for some , but it does not divide ? We answer this question in the positive when the are prime powers and and are subjected to certain restrictions.We use the result to prove that, if and is a set of three or more primes that contains all prime divisors of any number of the form for...
Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality has infinitely many solutions in prime variables p₁, p₂, p₃.