Amicable pairs with few distinct prime factors

2016 ◽  
Vol 12 (07) ◽  
pp. 1725-1732
Author(s):  
Florian Luca ◽  
M. Tip Phaovibul
Keyword(s):  
Prime Factors ◽  
Positive Integers ◽  
Sum Of Divisors

An amicable pair [Formula: see text] is a pair of distinct positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text] is the sum of divisors function. In this note, we prove that if [Formula: see text] are amicable and [Formula: see text] is odd, then [Formula: see text], where [Formula: see text] is the number of distinct prime factors.

scholarly journals ON A VARIATION OF A CONGRUENCE OF SUBBARAO

2012 ◽  
Vol 93 (1-2) ◽  
pp. 85-90 ◽  
Author(s):  
ANDREJ DUJELLA ◽  
FLORIAN LUCA
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Euler Function ◽  
Prime Factors ◽  
Finite Set ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractWe study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

scholarly journals Integers free of small prime factors in arithmetic progressions

2000 ◽  
Vol 157 ◽  
pp. 103-127 ◽  
Author(s):  
Ti Zuo Xuan
Keyword(s):  
Prime Number ◽  
Short Range ◽  
Error Term ◽  
Real Zero ◽  
Prime Number Theorem ◽  
Arithmetic Progressions ◽  
Real Character ◽  
Dickman Function ◽  
Prime Factors ◽  
Positive Integers

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

When the sieve works II

2020 ◽  
Vol 2020 (763) ◽  
pp. 1-24
Author(s):  
Kaisa Matomäki ◽  
Xuancheng Shao
Keyword(s):  
Expected Value ◽  
Constant Factor ◽  
Duke Math ◽  
Main Conjecture ◽  
Prime Factors ◽  
Positive Integers

AbstractFor a set of primes {\mathcal{P}}, let {\Psi(x;\mathcal{P})} be the number of positive integers {n\leq x} all of whose prime factors lie in {\mathcal{P}}. In this paper we classify the sets of primes {\mathcal{P}} such that {\Psi(x;\mathcal{P})} is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matomäki [A. Granville, D. Koukoulopoulos and K. Matomäki, When the sieve works, Duke Math. J. 164 2015, 10, 1935–1969] and their main conjecture is proved in this paper. In particular, our main theorem implies that, if not too many large primes are sieved out in the sense that\sum_{\begin{subarray}{c}p\in\mathcal{P}\\ x^{1/v}<p\leq x^{1/u}\end{subarray}}\frac{1}{p}\geq\frac{1+\varepsilon}{u},for some {\varepsilon>0} and {v\geq u\geq 1}, then\Psi(x;\mathcal{P})\gg_{\varepsilon,v}x\prod_{\begin{subarray}{c}p\leq x\\ p\notin\mathcal{P}\end{subarray}}\bigg{(}1-\frac{1}{p}\bigg{)}.

A note on the diophantine equation (xm − l) / (x − 1) = yn + l

1994 ◽  
Vol 116 (3) ◽  
pp. 385-389 ◽  
Author(s):  
Le Maohua
Keyword(s):  
Diophantine Equation ◽  
Rational Numbers ◽  
Prime Factors ◽  
Image Position ◽  
Positive Integers

Let ℤ, ℕ, ℚ denote the sets of integers, positive integers and rational numbers, respectively. Solutions (x, y, m, n) of the equation (1) have been investigated in many papers:Let ω(m), ρ(m) denote the number of distinct prime factors and the greatest square free factor of m, respectively. In this note we prove the following results.

scholarly journals WHEN THE SUM OF ALIQUOTS DIVIDES THE TOTIENT

2007 ◽  
Vol 50 (3) ◽  
pp. 563-569
Author(s):  
William D. Banks ◽  
Florian Luca
Keyword(s):  
Euler Function ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractLet $\varphi(\cdot)$ be the Euler function and let $\sigma(\cdot)$ be the sum-of-divisors function. In this note, we bound the number of positive integers $n\le x$ with the property that $s(n)=\sigma(n)-n$ divides $\varphi(n)$.

scholarly journals A variance method in combinatorial number theory

1969 ◽  
Vol 10 (2) ◽  
pp. 126-129 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Number Theory ◽  
Variance Method ◽  
Combinatorial Number Theory ◽  
Prime Factors ◽  
Integer Solutions ◽  
Image Position ◽  
Positive Integers

Let s = s(a1, a2,...., ar) denote the number of integer solutions of the equationsubject to the conditionsthe ai being given positive integers, and square brackets denoting the integral part. Clearly s (a1,..., ar) is also the number s = s(m) of divisors of which contain exactly λ prime factors counted according to multiplicity, and is therefore, as is proved in [1], the cardinality of the largest possible set of divisors of m, no one of which divides another.

scholarly journals The degree of primitive sequences and Erdős conjecture

2021 ◽  
Vol 52 ◽  
pp. 37-42
Author(s):  
Ilias Laib
Keyword(s):  
Prime Factors ◽  
Primitive Sequences ◽  
Positive Integers ◽  
Erdös Conjecture

A sequence A of strictly positive integers is said to be primitive if none of its term divides another. Z. Zhang proved a result, conjectured by Erdős and Zhang in 1993, on the primitive sequences whose the number of the prime factors of its terms counted with multiplicity is at most 4. In this paper, we extend this result to the primitive sequences whose the number of the prime factors of its terms counted with multiplicity is at most 5.

Prime factors in generalised Fibonacci sequences - a question posed by Gill Hatch

2003 ◽  
Vol 87 (509) ◽  
pp. 203-208
Author(s):  
R. P. Burn
Keyword(s):  
Recurrence Relation ◽  
Fibonacci Sequence ◽  
Prime Factors ◽  
Fibonacci Sequences ◽  
Positive Integers

Those who enjoy number patterns will no doubt have had many hours of pleasure exploring the Fibonacci sequence to various moduli, and especially in recognising the regularity with which various prime factors occur in thesequence (see [1]). Gill Hatch’s question is whether the occurrence of prime factors in generalised Fibonacci sequences is similarly predictable. Generalised Fibonacci sequences (Gn), abbreviated to GF sequences, are sequences of positive integers derived from the recurrence relation tn + 2 = tn + 1 + tn. In the case of the Fibonacci sequence (Fn), the first two terms are 1 and 1.

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