For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|nd2. On the other hand, we prove that for the function f(n) := ∑p|np2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.
Balog, A., On triplets with descending largest prime factors, Studia Sci. Math. Hungar., 38 (2001), 45–50.
A B., 'On triplets with descending largest prime factors' (2001) 38Studia Sci. Math. Hungar.: 45-50.
A B.On triplets with descending largest prime factorsStudia Sci. Math. Hungar.2001384550)| false