On the Number of Factorizations of an Integer

Mathematica Slovaca, 2018

Given an additive function f and a multiplicative function g, let E(f, g;x) = #{n ≤ x: f(n) = g(n)}. We study the size of E(ω,g;x) and E(Ω,g;x), where ω(n) stands for the number of distinct prime factors of n and Ω(n) stands for the number of prime factors of n counting multiplicity. In particular, we show that E(ω,g;x) and E(Ω,g;x) are $\begin{array}{} \displaystyle O\left(\frac{x}{\sqrt{\log\log x}}\right) \end{array}$ for any integer valued multiplicative function g. This improves an earlier result of De Koninck, Doyon and Letendre.

Transactions of the American Mathematical Society, 1986

The number Ψ ( x , y ) \Psi (x,y) of integers ≤ x \leq x and free of prime factors > y > y has been given satisfactory estimates in the regions y ≤ ( log ⁡ x ) 3 / 4 − ε y \leq {(\log x)^{3/4 - \varepsilon }} and y > exp ⁡ { ( log ⁡ log ⁡ x ) 5 / 3 + ε } y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\} . In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates Ψ ( x , y ) \Psi (x,y) uniformly for x ≥ y ≥ 2 x \geq y \geq 2 within a factor 1 + O ( ( log ⁡ y ) / ( log ⁡ x ) + ( log ⁡ y ) / y ) 1 + O((\log y)/(\log x) + (\log y)/y) . As an application, we derive a simple formula for Ψ ( c x , y ) / Ψ ( x , y ) \Psi (cx,y)/\Psi (x,y) , where 1 ≤ c ≤ y 1 \leq c \leq y . We also prove a short interval estimate for Ψ ( x , y ) \Psi (x,y) .

Given k, ∈ N + , let x i,j be, for 1 ≤ i ≤ k and 0 ≤ j ≤ , some fixed integers. Then, define sn := k i=1 j=0 x n j i,j for every n ∈ N +. We prove (by elementary means) that there exist infinitely many n for which the number of distinct prime factors of sn is greater than the super-logarithm of n to base θ, for some real constant θ > 1, if and only if there do not exist non-zero integers a 0 , b 0 ,. .. , a , b such that s 2n = i=0 a (2n) i i and s 2n−1 = i=0 b (2n−1) i i for all n. (In fact, we obtain a slightly more general result in the same spirit, where the numbers x i,j are rational.) In particular, for c 1 , x 1 ,. .. , c k , x k ∈ N + the number of distinct prime factors of the sum c 1 x n 1 + · · · + c k x n k is bounded, as n ranges over N + , if and only if x 1 = · · · = x k .

1996

International Journal of Mathematics and Mathematical Sciences, 2003

For each integern≥2, letP(n)denote its largest prime factor. LetS:={n≥2:ndoes not divideP(n)!}andS(x):=#{n≤x:n∈S}. Erdős (1991) conjectured thatSis a set of zero density. This was proved by Kastanas (1994) who established thatS(x)=O(x/logx). Recently, Akbik (1999) proved thatS(x)=O(x exp{−(1/4)logx}). In this paper, we show thatS(x)=x exp{−(2+o(1))×log x log log x}. We also investigate small and large gaps among the elements ofSand state some conjectures.

2017

We examine how closely a multiplicative function resembles an additive function. Given a multiplicative function $g$ and an additive function $f$, we examine the size of the quantity $E(f,g;x)=\# \{n\leq x:f(n)=g(n)\}$. We establish a lower bound for $E(\Omega,g,x)$.

Asian Research Journal of Mathematics

A factoriangular number is a sum of a factorial and its corresponding triangular number. This paper presents some forms of the generalization of factoriangular numbers. One generalization is the \(n^{(m)}\) -factoriangular number which is of the form \((n!)^{m}\) + \(S_m(n)\), where \((n!)^{m}\) is the \(m\)th power of the factorial of \(n\) and \(S_m(n)\) is the sum of the \(m\)-powers of \(n\). This generalized form is explored for the different values of the natural number \(m\). The investigation results to some interesting proofs of theorems related thereto. Two important formulas were generated for \((n)^{m}\) -factoriangular number: \(Ft_{n^{(m)}}\) = \(Ft_{n^{(2k)}}\) = \((n!)^{2k}\) + \(2n+1\over2k+1\)\([n^{2k-2}+P(n^{2k-3})]T_n\) for even \(m=2k\), and \(Ft_{n^{(m)}}\) = \(Ft_{n^{(2k+1)}}\) = \((n!)^{2k+1}\) + \(n(n+1)\over k+1\)\([n^{2k-2}+P(n^{2k-3})]T_n\) for odd \(m=2k+1\)

Canadian Journal of Mathematics, 1992

Let h: [0,1] → R be such that and define .In 1966, Erdős [8] proved that holds for almost all n, which by using a simple argument implies that in the case h(u) = u, for almost all n, He further obtained that, for every z > 0 and almost all n, and that where ϕ, ψ, are continuous distribution functions. Several other results concerning the normal growth of prime factors of integers were obtained by Galambos [10], [11] and by De Koninck and Galambos [6]. Let χ = ﹛xm : w ∈ N﹜ be a sequence of real numbers such that limm→∞ xm = +∞. For each x ∈ χ let be a set of primes p ≤x. Denote by p(n) the smallest prime factor of n. In this paper, we investigate the number of prime divisors p of n, belonging to for which Th(n,p) > z. Given Δ < 1, we study the behaviour of the function We also investigate the two functions , where, in each case, h belongs to a large class of functions.

Acta Arithmetica, 2013

Journal of Number Theory, 1997

It is shown that, for any k, there exist infinitely many positive integers n such that in the prime power factorization of n!, all first k primes appear to even exponents. This answers a question of Erdo s and Graham (``Old and New Problems and Results in Combinatorial Number Theory,'' L'Enseignement Mathe matique, Imprimerie Kundia, Geneva, 1980). A few generalizations are provided as well. 1997 Academic Press Question. Does there exist, for every fixed k, some n>1 with all the exponents _ 1 (n), _ 2 (n), ..., _ k (n) even? Our first result answers this question in the affirmative.