Total Number Of Prime Factors Research Articles

Research on the Total Number of Prime Factors of Odd Perfect Numbers not dividing 3, 5, 7, and 11

N이 홀수 완전수일 때 Ω(N)을 N을 소인수분해 했을 때 소인수의 지수의 합이라 하자. Hare (2007)에서 3, 5, 7, 11을 약수로 갖지 않는 홀수 완전수 N에 대해 Ω(N) ≥ 81을 증명하였다. 이 논문에서는 Hare (2007)에서 쓰인 아이디어를 개선하여 3, 5, 7, 11을 약수로 갖지 않는 홀수 완전수 N에 대해 Ω(N)≥ 117임을 보였다. 참고로 최근에 Ochem 외(2012)에서 모든 홀수완전수 N에 대해 Ω(N) ≥ 101임을 보였다.

Oscillations in weighted arithmetic sums

We examine oscillations in a number of sums of arithmetic functions involving [Formula: see text], the total number of prime factors of [Formula: see text], and [Formula: see text], the number of distinct prime factors of [Formula: see text]. In particular, we examine oscillations in [Formula: see text] and in [Formula: see text] for [Formula: see text], and in [Formula: see text]. We show for example that each of the inequalities [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] is true infinitely often, disproving some hypotheses of Sun.

Consecutive Integers with Close Kernels

AbstractLet $k$ be an arbitrary positive integer and let $\unicode[STIX]{x1D6FE}(n)$ stand for the product of the distinct prime factors of $n$. For each integer $n\geqslant 2$, let $a_{n}$ and $b_{n}$ stand respectively for the maximum and the minimum of the $k$ integers $\unicode[STIX]{x1D6FE}(n+1),\unicode[STIX]{x1D6FE}(n+2),\ldots ,\unicode[STIX]{x1D6FE}(n+k)$. We show that $\liminf _{n\rightarrow \infty }a_{n}/b_{n}=1$. We also prove that the same result holds in the case of the Euler function and the sum of the divisors function, as well as the functions $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, which stand respectively for the number of distinct prime factors of $n$ and the total number of prime factors of $n$ counting their multiplicity.

On a pair of zeta functions

Let [Formula: see text] be a positive integer, and define [Formula: see text] for [Formula: see text], where [Formula: see text] denotes the number of distinct prime factors of [Formula: see text], and [Formula: see text] represents the total number of prime factors of [Formula: see text] (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that [Formula: see text] which is similar to the known identity [Formula: see text] equivalent to the Prime Number Theorem. For [Formula: see text], we prove that [Formula: see text] We also raise a hypothesis on the parities of [Formula: see text] which implies the Riemann Hypothesis.

On a ternary Diophantine inequality

Let \(\Omega (n)\) be the total number of prime factors of n, and let \(\lambda _j\) be the real numbers satisfying suitable conditions. Let \(J_k(N)\) denote the number of solutions to the inequality $$\begin{aligned} \left\{ \begin{array}{ll} |\lambda _1n_1+\lambda _2n_2+\lambda _3n_3+\eta |<\varepsilon ,&{} \\ \Omega (n_j)= k,&{} \\ 2\le n_j\le N \quad (j=1,2,3).&{} \end{array} \right. \end{aligned}$$ In this note, we investigate the properties of \(J_k(N)\) for any integer \(k\ge 1\), which is allowed to tend to infinity with respect to N. Using an asymptotic formula for the weighted exponential sums, we obtain a sharper lower bound for it and also discuss an application of the main result.

A generalization of the Montgomery–Hooley theorem

Let Ω(n) be the total number of prime factors of n andΘk(x;q,a)=∑n≤xn≡a(modq)Ω(n)=k1, where k is allowed to tend to infinity with respect to x. Combining the circle method with the Selberg–Delange method, together with the result of Bombieri-type sum for exponential sums, we investigate the behavior of the error term of Θk(x;q,a) in the “mean” and obtain its upper bound.

On the normal number of prime factors of φ(n) subject to certain congruence conditions

Let q≥2 be an integer and Sq(n) denote the sum of the digits in base q of the positive integer n. It is proved that for every real number α and β with α<β,limx⟶+∞⁡1x♯{n≤x:α≤v(φ(n))−12b(log⁡log⁡n)213b(log⁡log⁡n)32≤β}=12π∫αβe−t22dt, where v(n) is either ω˜(n) or Ω˜(n), the number of distinct prime factors and the total number of prime factors p of a positive integer n such that Sq(p)≡amodb (a,b∈Z, b≥2). This extends the results known through the work of P. Erdős and C. Pomerance, M.R. Murty and V.K. Murty to primes under digital constraint.

On the number of prime factors of an odd perfect number

Let ω(n) and Ω(n) denote, respectively, the total number of prime factors and the number of distinct prime factors of the integer n. Euler proved that an odd perfect number N is of the form N = pᶱm² where p ≡ e ≡ 1 (mod 4), p is prime, and p ∤ m. This implies that Ω(N) ≥ 2ω(N) − 1. We. We prove that Ω(N) ≥ (18ω(N) −31) / 7andΩ(N) ≥ 2ω(N) + 51.

On long κ-tuples with few prime factors

We prove that there are infinitely many integers n such that the total number of prime factors of (n+h1)…(n+hκ) is exactly (1 + o(1))κ Log k. Our result even ensures us that these prime factors are fairly evenly distributed among and factor n+hi.

On strings of consecutive integers with a distinct number of prime factors

Let ω ( n ) \omega (n) be the number of distinct prime factors of n n . For any positive integer k k let n = n k n=n_k be the smallest positive integer such that ω ( n + 1 ) , … , ω ( n + k ) \omega (n+1),\ldots ,\omega (n+k) are mutually distinct. In this paper, we give upper and lower bounds for n k n_k . We study the same quantity when ω ( n ) \omega (n) is replaced by Ω ( n ) \Omega (n) , the total number of prime factors of n n counted with repetitions.

New techniques for bounds on the total number of prime factors of an odd perfect number

Let �(n) denote the sum of the positive divisors of n. We say that n is perfect if �(n) = 2n. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must

On almost prime k-tuples

Let k ⩾ 2 and a i , b i ( 1 ⩽ i ⩽ k ) be integers such that a i > 0 and ∏ 1 ⩽ i < j ⩽ k ( a i b j − a j b i ) ≠ 0 . Let Ω ( m ) denote the total number of prime factors of m. Suppose ∏ ( n ) : = ∏ i = 1 k ( a i n + b i ) has no fixed prime divisors. Results of the form # { n ⩽ x : Ω ( ∏ ( n ) ) ⩽ r k } ≫ x ( log x ) − k where r k is asymptotic to k log k have been obtained by using sieve methods, in particular weighted sieves. In this paper, we use another kind of weighted sieve due to Selberg to obtain improved admissible values for r k .

Hypercyclic and chaotic convolution operators associated with the Dunkl operator on &lt;Emphasis Type=?Bold?&gt;C&lt;/Emphasis&gt;

We show that there exist infinitely many positive integers m not of the form n -o( n ) for any positive integer n . Here, o( n ) stands for the number of distinct prime factors of n . A similar result holds with o( n ) replaced by the total number of prime factors of n (counting multiplicities), or by the number of divisors of n .

More on the total number of prime factors of an odd perfect number

Let σ ( n ) \sigma (n) denote the sum of the positive divisors of n n . We say that n n is perfect if σ ( n ) = 2 n \sigma (n) = 2 n . Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form N = p α ∏ j = 1 k q j 2 β j N = p^\alpha \prod _{j=1}^k q_j^{2 \beta _j} , where p , q 1 , … , q k p, q_1, \ldots , q_k are distinct primes and p ≡ α ≡ 1 ( mod 4 ) p \equiv \alpha \equiv 1 \pmod {4} . Define the total number of prime factors of N N as Ω ( N ) := α + 2 ∑ j = 1 k β j \Omega (N) := \alpha + 2 \sum _{j=1}^k \beta _j . Sayers showed that Ω ( N ) ≥ 29 \Omega (N) \geq 29 . This was later extended by Iannucci and Sorli to show that Ω ( N ) ≥ 37 \Omega (N) \geq 37 . This paper extends these results to show that Ω ( N ) ≥ 47 \Omega (N) \geq 47 .

On the total number of prime factors of an odd perfect number

We say n ∈ N is perfect if σ(n) = 2n, where σ(n) denotes the sum of the positive divisors of n. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form n = pα Φj=1k qj2βj, where p,q1,..., qk are distinct primes and p ≡ α ≡ 1 (mod 4). We prove that if βj ≡ 1 (mod 3) or βj ≡ 2 (mod 5) for all j, 1 ≤ j ≤ k, then 3|n. We also prove as our main result that Ω(n) ≥ 37, where Ω(n) = α + 2 Σj=1k βj. This improves a result of Sayers (Ω(n) ≥ 29) given in 1986.

Sur la répartition des valeurs des fonctions arithmétiques. Le nombre de facteurs premiers d'un entier

This paper is concerned with the quantityN(x, m), the number of positive integersn, 1⩽n⩽x, for whichΩ(n)=m, whereΩ:(n) denotes the total number of prime factors (counted with multiplicities) ofn. The main purpose of this article is to present three powerful analytic methods, due, respectively, to Selberg, van der Waerden, and the author, the combination of which allows one to completely characterize the asymptotic behavior of the quantityN(x, m) as the second parameter varies through all its possible values, namely 1⩽m⩽(log x)/log 2. These methods constitute, in a certain sense, a compact and effective set of analytical tools and apply also to the distribution function associated withn(x, m). All these methods are quite general and applicable to many other arithmetic functions.

The average prime divisor of an integer in short intervals

and as usual co(n) is the number of distinct prime factors of n, f2(n) is the total number of prime factors of n, f II n means that p~ (p prime) exactly divides n. It was shown in [1] that, for each fixed natural number m, there exist computable constants el, e z, ..., era, fa,f2 ..... fro, 0 2). These questions were considered in a general setting by De Koninck and Mercier [2], where under suitable conditions asymptotic relations of the form (1.3) Y~ f(P(n)) = (1 + o(1)) Z f(n) (x ~co)

Sums of Quotients of Additive Functions

Denote by $\omega (n)$ and $\Omega (n)$ the number of distinct prime factors of $n$ and the total number of prime factors of $n$, respectively. Given any positive integer $\alpha$, we prove that \[ \sum \limits _{2 \leqq n \leqq x} {\Omega (n)/\omega } (n) = x + x\sum \limits _{i = 1}^\alpha {{a_i}/{{(\log \log x)}^i} + O{{(x/\log \log x)}^{\alpha + 1}}),} \] where ${a_1} = \sum \nolimits _p {1/p(p - 1)}$ and all the other ${a_i}$’s are computable constants. This improves a previous result of R. L. Duncan.

Sums of quotients of additive functions

Denote by ω ( n ) \omega (n) and Ω ( n ) \Omega (n) the number of distinct prime factors of n n and the total number of prime factors of n n , respectively. Given any positive integer α \alpha , we prove that \[ ∑ 2 ≦ n ≦ x Ω ( n ) / ω ( n ) = x + x ∑ i = 1 α a i / ( log ⁡ log ⁡ x ) i + O ( x / log ⁡ log ⁡ x ) α + 1 ) , \sum \limits _{2 \leqq n \leqq x} {\Omega (n)/\omega } (n) = x + x\sum \limits _{i = 1}^\alpha {{a_i}/{{(\log \log x)}^i} + O{{(x/\log \log x)}^{\alpha + 1}}),} \] where a 1 = ∑ p 1 / p ( p − 1 ) {a_1} = \sum \nolimits _p {1/p(p - 1)} and all the other a i {a_i} ’s are computable constants. This improves a previous result of R. L. Duncan.