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1
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On the unimodal character of the frequency function of the largest prime factor

100%
De Koninck J.-M., Sweeney J.
Colloquium Mathematicum
|
2001
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tom 88
|
nr 2
159-174
EN
The main objective of this paper is to analyze the unimodal character of the frequency function of the largest prime factor. To do that, let P(n) stand for the largest prime factor of n. Then define f(x,p): = #{n ≤ x | P(n) = p}. If f(x,p) is considered as a function of p, for 2 ≤ p ≤ x, the primes in the interval [2,x] belong to three intervals I₁(x) = [2,v(x)], I₂(x) = ]v(x),w(x)[ and I₃(x) = [w(x),x], with v(x) < w(x), such that f(x,p) increases for p ∈ I₁(x), reaches its maximum value in I₂(x), in which interval it oscillates, and finally decreases for p ∈ I₃(x). In fact, we show that v(x) ≥ √(log x) and w(x) ≤ √x. We also provide several conditions on primes p ≤ q so that f(x,p) ≥ f(x,q).
2
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On the composition of the Euler function and the sum of divisors function

100%
De Koninck J.-M., Luca F.
Colloquium Mathematicum
|
2007
|
tom 108
|
nr 1
31-51
EN
Let H(n) = σ(ϕ(n))/ϕ(σ(n)), where ϕ(n) is Euler's function and σ(n) stands for the sum of the positive divisors of n. We obtain the maximal and minimal orders of H(n) as well as its average order, and we also prove two density theorems. In particular, we answer a question raised by Golomb.
3
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Exponential sums involving the largest prime factor function

100%
De Koninck J.-M., Kátai I.
Acta Arithmetica
|
2011
|
tom 146
|
nr 3
233-245
4
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On the distribution of subsets of primes in the prime factorization of integers

100%
De Koninck J.-M., Kátai I.
Acta Arithmetica
|
1995
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tom 72
|
nr 2
169-200
5
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Normal numbers and the middle prime factor of an integer

100%
De Koninck J.-M., Kátai I.
Colloquium Mathematicum
|
2014
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tom 135
|
nr 1
69-77
EN
Let pₘ(n) stand for the middle prime factor of the integer n ≥ 2. We first establish that the size of log pₘ(n) is close to √(log n) for almost all n. We then show how one can use the successive values of pₘ(n) to generate a normal number in any given base D ≥ 2. Finally, we study the behavior of exponential sums involving the middle prime factor function.
6
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Shifted values of the largest prime factor function and its average value in short intervals

100%
De Koninck J.-M., Kátai I.
Colloquium Mathematicum
|
2016
|
tom 143
|
nr 1
39-62
EN
We obtain estimates for the average value of the largest prime factor P(n) in short intervals [x,x+y] and of h(P(n)+1), where h is a complex-valued additive function or multiplicative function satisfying certain conditions. Letting $s_{q}(n)$ stand for the sum of the digits of n in base q ≥ 2, we show that if α is an irrational number, then the sequence $(αs_{q}(P(n)))_{n∈ ℕ}$ is uniformly distributed modulo 1.
7
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Sur la proximité des nombres puissants

100%
De Koninck J.-M., Luca F.
Acta Arithmetica
|
2004
|
tom 114
|
nr 2
149-157
8
Content available remote

Sur la quantité de nombres économiques

81%
De Koninck J.-M., Doyon N., Luca F.
Acta Arithmetica
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2007
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tom 127
|
nr 2
125-143
9
Content available remote

On the counting function for the Niven numbers

81%
De Koninck J.-M., Doyon N., Kátai I.
Acta Arithmetica
|
2003
|
tom 106
|
nr 3
265-275
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