scholarly journals On a sum involving the number of distinct prime factors function related to the integer part function

2020 ◽  
Vol 26 (4) ◽  
pp. 52-56
Author(s):  
Mihoub Bouderbala ◽  
◽  
Meselem Karras ◽  
Keyword(s):  
Asymptotic Formula ◽  
Integer Part ◽  
Prime Divisors ◽  
Prime Factors

In this paper, we obtain asymptotic formula on the sum \sum\limits_{n\leq x}\omega \left( \left\lfloor \frac{x}{n}\right\rfloor \right) , where \omega \left( n\right) denote the number of distinct prime divisors of n and \left\lfloor t\right\rfloor denotes the integer part of t.

scholarly journals Orders of reductions of elliptic curves with many and few prime factors

2017 ◽  
Vol 13 (08) ◽  
pp. 2115-2134
Author(s):  
Lee Troupe
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Extreme Values ◽  
Complex Multiplication ◽  
Prime Divisors ◽  
Prime Factors

In this paper, we investigate extreme values of [Formula: see text], where [Formula: see text] is an elliptic curve with complex multiplication and [Formula: see text] is the number-of-distinct-prime-divisors function. For fixed [Formula: see text], we prove an asymptotic formula for the quantity [Formula: see text]. The same result holds for the quantity [Formula: see text] when [Formula: see text]. This asymptotic formula matches what one might expect, based on a result of Delange concerning extreme values of [Formula: see text]. The argument is worked out in detail for the curve [Formula: see text], and we discuss how the method can be adapted for other CM elliptic curves.

On the Normal Growth of Prime Factors of Integers

1992 ◽  
Vol 44 (6) ◽  
pp. 1121-1154 ◽  
Author(s):  
J. M. De Koninck ◽  
I. Kátai ◽  
A. Mercier
Keyword(s):  
Prime Factor ◽  
Continuous Distribution ◽  
Normal Growth ◽  
Distribution Functions ◽  
Prime Divisors ◽  
Simple Argument ◽  
Prime Factors ◽  
Almost All ◽  
Number Of Prime Divisors ◽  
Class Of Functions

AbstractLet 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.

A Variation on the Algorithm for GCD and LCM

1982 ◽  
Vol 30 (3) ◽  
pp. 46
Author(s):  
Verna M. Adams
Keyword(s):  
Prime Numbers ◽  
Prime Divisors ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Prime Factors

An algorithm sometimes presented for finding the least common multiple (LCM) of two numbers uses tbe technique of simultaneously finding the prime factors of the numbers. This technique is shown in figure 1. Both numbers are checked for divisibility by 2, then by 3, by 5, and so on. If the divisor does not divide one of the numbers, the number is written on the next line as shown in steps 4 and 5. This process continues until all numbers to the left and on the bottom are prime numbers, or it can be continued, as shown in figure 1, until the numbers across the bottom are all ones. The least common multiple is the product of all of the prime divisors. Thus, LCM (80, 72) = 24 · 32 · 5.

ON THE MEAN VALUE OF THE INDEX OF COMPOSITION OF AN INTEGER II

2012 ◽  
Vol 09 (02) ◽  
pp. 431-445 ◽  
Author(s):  
DEYU ZHANG ◽  
MEIMEI LÜ ◽  
WENGUANG ZHAI
Keyword(s):  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Error Term ◽  
Mean Value ◽  
Prime Factors ◽  
The Mean

For each integer n ≥ 2, let [Formula: see text] be the index of composition of n, where γ(n) ≔ ∏p∣np. The index of composition of an integer measures the multiplicity of its prime factors. In this paper, we obtain a new asymptotic formula of the sum ∑n≤xλ-k(n). Furthermore, we improve the error term under the Riemann Hypothesis.

A note on weakly prime-additive numbers

2021 ◽  
pp. 1-4
Author(s):  
Jin-Hui Fang
Keyword(s):  
Positive Integer ◽  
Prime Divisors ◽  
Prime Factors ◽  
Positive Integers

A positive integer [Formula: see text] is called weakly prime-additive if [Formula: see text] has at least two distinct prime divisors and there exist distinct prime divisors [Formula: see text] of [Formula: see text] and positive integers [Formula: see text] such that [Formula: see text]. It is easy to see that [Formula: see text]. In this paper, intrigued by De Koninck and Luca’s work, we further determine all weakly prime-additive numbers [Formula: see text] such that [Formula: see text], where [Formula: see text] are distinct odd prime factors of [Formula: see text].

scholarly journals Sieving Positive Integers by Primes

2019 ◽  
Vol 11 (1) ◽  
pp. 17
Author(s):  
Samir Brahim Behaouari
Keyword(s):  
Asymptotic Formula ◽  
Prime Factors ◽  
Positive Integers

Let Q be a set of primes and let &Psi;(x, y, Q) be the number of positive integers less than or equal to x that have no prime factors from Q exceeding the integer y. We have enhanced an asymptotic formula for &Psi;(x; y; Q) after highlighting some properties of the function &Psi;.

scholarly journals A remark on the number of distinct prime divisors of integers

2017 ◽  
Vol 48 (1) ◽  
pp. 13-15
Author(s):  
Mehdi Hassani
Keyword(s):  
Asymptotic Formula ◽  
Prime Divisors ◽  
Related Sequence ◽  
Curve Patterns

We study the asymptotic formula for the sum $\sum_{n\leqslant x}\o(n)$ where $\o(n)$ denotes the number of distinct prime divisors of $n$, and we perform some computations which detect curve patterns in the distribution of a related sequence.

scholarly journals Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields Whose Discriminant Has Only Two Prime Factors

2012 ◽  
Vol 2012 ◽  
pp. 1-4
Author(s):  
A. Pekin
Keyword(s):  
Quadratic Fields ◽  
Class Numbers ◽  
Sufficient Condition ◽  
Prime Divisors ◽  
Imaginary Quadratic Fields ◽  
Prime Factors

We will prove a theorem providing sufficient condition for the divisibility of class numbers of certain imaginary quadratic fields by 2g, whereg>1is an integer and the discriminant of such fields has only two prime divisors.

PRIME DIVISORS ARE POISSON DISTRIBUTED

2007 ◽  
Vol 03 (01) ◽  
pp. 1-18 ◽  
Author(s):  
ANDREW GRANVILLE
Keyword(s):  
Prime Divisors ◽  
Prime Factors ◽  
Cycle Lengths ◽  
Almost All

We show that the set of prime factors of almost all integers are "Poisson distributed", and that this remains true (appropriately formulated) even when we restrict the number of prime factors of the integer. Our results have inspired analogous results about the distribution of cycle lengths of permutations.

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