Let γ(n) stand for the product of the prime factors of n. The index of composition λ(n) of an integer n ≥ 2 is defined as λ(n) = log n/ log γ(n) with λ(1) = 1. Given an arbitrary integer k ≥ 0 and letting σk(n) be the k-fold iterate of the sum of divisors function, we show that, given any real number ε > 0, λ(σk(n)) < 1 + ε for almost all positive integers n.

let $cr_{n}(f)$ denote the algebra of $n times n$ circulant matrices over the field $f$. in this paper, we study the unit group of $cr_{n}(f_{p^m})$, where $f_{p^m}$ denotes the galois field of order $p^{m}$, $p$ prime.

the energy minima of systems made of an alanine molecule, a lithium cation and a various number of watermolecules have been determined with the help of quantum mechanical computations at the b3lyp densityfunctional theory level of computation and the standard 6-311++g (d,p) basis set. several structures, close inenergy are found and the presence of one or two water molecules around the cation m...

• An arithmetic function takes positive integers as inputs and produces real or complex numbers as outputs. • If f is an arithmetic function, the divisor sum Df(n) is the sum of the values of f at the positive divisors of n. • τ (n) is the number of positive divisors of n; σ(n) is the sum of the positive divisors of n. • The Möbius function μ(n) is 1 if n = 1 and 0 if n has a repeated prime fac...

Let R be a prime ring of char R = 2, d a nonzero derivation of R, ρ a nonzero right ideal of R and 0 = b ∈ R such that b[[d2[x, y], [x, y]]n = 0 for all x, y ∈ ρ, n ≥ 1 fixed integer. If [ρ, ρ]ρ = 0 then either bρ = 0 or d(ρ)ρ = 0. Mathematics Subject Classification: 16W25, 16R50, 16N60

Conjecture 1 (1988-04-23). Let a0, . . . , an−1, b0, . . . , bn−1 ∈ N. Suppose that ∑n−1 r=0 are 2πir/n = ∑n−1 r=0 bre , and that the least prime divisor p = p(n) of n is greater than |{0 6 r < n : ar 6= 0}| and |{0 6 r < n : br 6= 0}|. Then ar = br for all r ∈ R(n) = {0, 1, . . . , n− 1}. Remark 1. M. Newman [Math. Ann. 1971] showed that if c0, . . . , cn−1 ∈ Q, ∑n−1 r=0 cre 2πir/n = 0 and |{0...

For a prime p and nonnegative integers n, k, consider the set A (p) n,k = {x ∈ [0, 1, ..., n] : p|| ( n x ) }. Let the expansion of n + 1 in base p be n + 1 = α0p ν + α1p ν−1 + · · · + αν , where 0 ≤ αi ≤ p − 1, i = 0, . . . , ν. Then n is called a binomial coefficient predictor in base p(p-BCP), if |A (p) n,k| = αkp , k = 0, 1, . . . , ν. We give a full description of the p-BCP’s in every base p.

Here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. Also we find integers $n$ for which, these groups are $n$-central.