On relatively prime sets.
On relatively prime subsets and supersets.
On sets characterizing additive arithmetical functions, I
On sets characterizing additive arithmetical functions, I
On short sums of certain multiplicative functions.
On some aspects of multiplicative arithmetic functions
On some extensions of Jordan's arithmetic functions.
On some modifications of two theorems of Erdős
On some number-theoretic sums introduced by Jacobsthal
On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ
Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n)...
On some rings of arithmetical functions.
On square values of the product of the Euler totient and sum of divisors functions
If n is a positive integer such that ϕ(n)σ(n) = m² for some positive integer m, then m ≤ n. We put m = n-a and we study the positive integers a arising in this way.
On strings of consecutive economical numbers of arbitrary length.
On the arithmetical functions and .
On the Average Order of an Arithmetical Function
On the almost periodic behaviour of certain arithmetical convolutions
On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function.
On the composition of some arithmetic functions. II.
On the composition of the arithmetic functions σ and φ
On the composition of the Euler function and the sum of divisors function
Let H(n) = σ(ϕ(n))/ϕ(σ(n)), where ϕ(n) is Euler's function and σ(n) stands for the sum of the positive divisors of n. We obtain the maximal and minimal orders of H(n) as well as its average order, and we also prove two density theorems. In particular, we answer a question raised by Golomb.