On the prime factors of the iterates of the Ramanujan τ–function

2020 ◽  
Vol 63 (4) ◽  
pp. 1031-1047
Author(s):  
Florian Luca ◽  
Sibusiso Mabaso ◽  
Pantelimon Stănică
Keyword(s):  
Positive Integer ◽  
Prime Factors

AbstractIn this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri 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 On f(n) modulo Ω(n) and ω(n) when f is a polynomial

2004 ◽  
Vol 77 (2) ◽  
pp. 149-164 ◽  
Author(s):  
Florian Luca
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Density Zero ◽  
Prime Factors ◽  
Nonzero Polynomial

AbstractIn this paper we show that if f (X) ∈; Z [X ] is a nonzero polynomial, then ω(n)/f(n) holds only on a set of n of asymptotic density zero, where for a positive integer n the number ω(n) counts the number of distinct prime factors ofn.

scholarly journals The Solutions of Generalized Euler Function Equation φ_2(n-φ_2 (n))=2ω(n)

2021 ◽  
pp. 15-22
Author(s):  
Xu Yifan ◽  
Shen Zhongyan
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Euler Function ◽  
Prime Factors ◽  
Integer Solutions

By using the properties of Euler function, an upper bound of solutions of Euler function equation  is given, where  is a positive integer. By using the classification discussion and the upper bound we obtained, all positive integer solutions of the generalized Euler function equation  are given, where is the number of distinct prime factors of n.

scholarly journals Uniform congruence counting for Schottky semigroups in SL2(𝐙)

2019 ◽  
Vol 2019 (753) ◽  
pp. 89-135 ◽  
Author(s):  
Michael Magee ◽  
Hee Oh ◽  
Dale Winter
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Continued Fractions ◽  
Euclidean Norm ◽  
Limit Set ◽  
The Family ◽  
Prime Factors ◽  
Uniform Congruence

AbstractLet Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})}, and for {q\in\mathbf{N}}, let{\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}}be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls {B_{R}} in {M_{2}(\mathbf{R})} of radius R: for all positive integer q with no small prime factors,\#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(% \mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon})as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})}, which arises in the study of Zaremba’s conjecture on continued fractions.

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 ON THE SQUARE-FREE PARTS OF ⌊en!⌋

2007 ◽  
Vol 49 (2) ◽  
pp. 391-403
Author(s):  
FLORIAN LUCA ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Absolute Constant ◽  
Free Part ◽  
Prime Factors ◽  
Image Position

AbstractIn this note, we show that if we write ⌊en!⌋ = s(n)u(n)2, where s(n) is square-free then has at least C log log N distinct prime factors for some absolute constant C > 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes dividing the mth power-free part of s(n) as n ranges from 1 to N, where m ≥ 3 is a positive integer. As an application of such results, we give an upper bound on the number of n ≤ N such that ⌊en!⌋ is a square.

scholarly journals ON NEAR-PERFECT NUMBERS WITH TWO DISTINCT PRIME FACTORS

2013 ◽  
Vol 88 (3) ◽  
pp. 520-524 ◽  
Author(s):  
XIAO-ZHI REN ◽  
YONG-GAO CHEN
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Prime Factors ◽  
Perfect Numbers

AbstractRecently, Pollack and Shevelev [‘On perfect and near-perfect numbers’, J. Number Theory 132 (2012), 3037–3046] introduced the concept of near-perfect numbers. A positive integer $n$ is called near-perfect if it is the sum of all but one of its proper divisors. In this paper, we determine all near-perfect numbers with two distinct prime factors.

scholarly journals On Dirichlet's divisor problem

1931 ◽  
Vol 134 (823) ◽  
pp. 192-202 ◽  
Keyword(s):  
Positive Integer ◽  
Divisor Problem ◽  
Euler’S Constant ◽  
Number Of Divisors ◽  
Prime Factors ◽  
Euler's Constant

1. Let d ( n ) denote the number of divisors of the positive integer n , so that, if n = p 1 a 1 . . . p r ar is the canonical expression of n in prime factors, d ( n ) = (1 + a 1 ) . . . (1 + a r ), and let d ( x ) = 0 if x is not an iteger; then if (1. 1) D ( x ) = Σ' n ≤ x d ( n ) = Σ n ≤ x d ( n ) ─ ½ d ( x ), and (1. 2) Δ ( x ) = D ( x ) ─ x log x ─ (2C ─ 1) x ─ ¼, where C is Euler's constant, it was proved by Dirichlet in 1849 that (1. 21) Δ ( x ) = O (√ x ),

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