On a variant of Giuga numbers

Grau, J. M.; Luca, Florian; Oller-Marcén, Antonio M.

doi:10.1007/s10114-011-1148-7

Cited by 2 publications

(5 citation statements)

References 13 publications

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“…Alternatively we can write this as

δ (⋃_{j < k} G_{p_{j}}^{b}) = - \sum_{\begin{matrix} m > 1, m ∣ p_{2} p_{3} \dots p_{k - 1} \\ \gcd (m, φ (m) ∣ b \end{matrix}} \frac{μ (m)}{lcm (m, λ (m))} .

It is not difficult to see, cf. , that the series

\sum_{gcd (m, φ (m)) ∣ b} \frac{μ (m)}{lcm (m, λ (m))}

converges absolutely. Using this, and , we then obtain the following result.…”

Section: The Affine Casementioning

confidence: 78%

“…It is not difficult to see, cf. [4], that the series gcd(m,ϕ(m))|b μ(m) lcm(m, λ(m)) converges absolutely. Using this, (4.2) and (4.3), we then obtain the following result.…”

Section: Corollary 44 Putmentioning

confidence: 97%

“…The asymptotic density of

M_{f_{_{1, \pm 1}}}

is closely related to that of the set

P : = \{n \geq 1 : 2 ∤ n, S_{\frac{n - 1}{2}} \equiv 0 ((), \mod, n)\},

which was defined and studied in and where it is shown that

δ (frakturP) = \frac{1}{2} + \sum_{\frac{0pt}{m > 2}} \frac{{(- 1)}^{ω (m)}}{2 m λ (m)} \in [0.379005, 0.379826] .

By combining this with Corollary we reach the following conclusion. Proposition We have

δ (M_{f_{_{1, \pm 1}}}) = 2 δ (frakturP) - 1 / 4 \in [0.50801, 0.50966]

.…”

Section: The Affine Casementioning

confidence: 99%

“…where ω(m) is the number of distinct prime factors of m. The asymptotic density of M f 1,±1 is closely related to that of the set P := n ≥ 1 : 2 n, S n−1 2 ≡ 0 (mod n) , which was defined and studied in [4] and where it is shown that…”

Section: Corollary 47mentioning

confidence: 99%

“…Moreover, inspired by Giuga's ideas we investigate the congruence S f (n) (n) ≡ 0 (mod n) for some functions f . This work started in [4], where the case f (n) = (n − 1)/2 was considered. The case of arithmetic functions f such that p − 1 f ( p) for every prime p is of special interest.…”

Section: Introductionmentioning

confidence: 99%

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Solutions of the congruence

Grau

Moree

Oller-Marcén

2015

Mathematische Nachrichten

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“…Alternatively we can write this as

δ (⋃_{j < k} G_{p_{j}}^{b}) = - \sum_{\begin{matrix} m > 1, m ∣ p_{2} p_{3} \dots p_{k - 1} \\ \gcd (m, φ (m) ∣ b \end{matrix}} \frac{μ (m)}{lcm (m, λ (m))} .

It is not difficult to see, cf. , that the series

\sum_{gcd (m, φ (m)) ∣ b} \frac{μ (m)}{lcm (m, λ (m))}

converges absolutely. Using this, and , we then obtain the following result.…”

Section: The Affine Casementioning

confidence: 78%

“…It is not difficult to see, cf. [4], that the series gcd(m,ϕ(m))|b μ(m) lcm(m, λ(m)) converges absolutely. Using this, (4.2) and (4.3), we then obtain the following result.…”

Section: Corollary 44 Putmentioning

confidence: 97%

“…The asymptotic density of

M_{f_{_{1, \pm 1}}}

is closely related to that of the set

P : = \{n \geq 1 : 2 ∤ n, S_{\frac{n - 1}{2}} \equiv 0 ((), \mod, n)\},

which was defined and studied in and where it is shown that

δ (frakturP) = \frac{1}{2} + \sum_{\frac{0pt}{m > 2}} \frac{{(- 1)}^{ω (m)}}{2 m λ (m)} \in [0.379005, 0.379826] .

By combining this with Corollary we reach the following conclusion. Proposition We have

δ (M_{f_{_{1, \pm 1}}}) = 2 δ (frakturP) - 1 / 4 \in [0.50801, 0.50966]

.…”

Section: The Affine Casementioning

confidence: 99%

Section: Corollary 47mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solutions of the congruence

Grau

Moree

Oller-Marcén

2015

Mathematische Nachrichten

Self Cite

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Variations on Giuga Numbers and Giuga’s Congruence

Grau

Oller-Marcén

2016

Ukr Math J

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A k-strong Giuga number is a composite integer such that n−1 j=1 j n−1 ≡ −1 (mod n). We consider the congruence n−1 j=1 j k(n−1) ≡ −1 (mod n) for each k ∈ N (thus extending Giuga's ideas for k = 1). In particular, it is proved that a pair (n, k) with composite n satisfies this congruence if and only if n is a Giuga number and λ(n) | k(n − 1). In passing, we establish some new characterizations of Giuga numbers and study some properties of the numbers n satisfying λ(n) | k(n − 1). k-Сильне число Гюга-це складене цiле число таке, що n−1 j=1 j n−1 ≡ −1 (mod n). Ми розглядаємо конгруенцiю n−1 j=1 j k(n−1) ≡ −1 (mod n) для кожного k ∈ N (таким чином ми розширюємо iдеї Гюга на випадок k = 1). Як частинний випадок доведено, що пара (n, k) зi складеним n задовольняє цю конгруенцiю тодi i тiльки тодi, коли n-число Гюга та λ(n) | k(n − 1). Крiм того, встановлено деякi новi характеристики чисел Гюга та вивчено властивостi чисел n, що задовольняють λ(n) | k(n − 1).

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On a variant of Giuga numbers

Abstract: In this paper, we characterize the odd positive integers n satisfying the congruence n−1 j=1 j n−1 2 ≡ 0 (mod n). We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8.

Cited by 2 publications

References 13 publications

Solutions of the congruence

Solutions of the congruence

Variations on Giuga Numbers and Giuga’s Congruence

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