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We study divisibility properties of a set { f 1 (U n (s)) , ... , f m (U n (s)) } , where f 1 , ... , f m are polynomials in s variables over Z and U n (s) is a point picked uniformly at random from the set { 1 , ... , n } s. We show that, as n → ∞ , the GCD and the suitably normalized LCM of this set converge in distribution to a.s. finite random variables under mild assumptions on f 1 , ... , f m. Our approach is based on the known fact that the uniform distribution on { 1 , ... , n } converges to the Haar measure on the ring Z ˆ of profinite integers, combined with the Lang–Weil bounds and tools from probability theory. [ABSTRACT FROM AUTHOR]

Let (ξ1, η1), (ξ2, η2), ... be independent identically distributed N²-valued random vectors with arbitrarily dependent components. The sequence (Θk)k∈N defined by Θk = Πk-1 ⋅ ηk, where Π0 = 1 and Πk = ξ1 ⋅ ... ⋅ ξk for k ∈ N, is called a multiplicative perturbed random walk. Arithmetic properties of the random sets {Π1,Π2, ..., Πk} ⊆ N and {Θ1,Θ2, ..., Θk} ⊆ N, k ∈ N, are studied. In particular, distributional limit theorems for their prime counts and for the least common multiple are derived. [ABSTRACT FROM AUTHOR]

Let $({\xi _{1}},{\eta _{1}})$, $({\xi _{2}},{\eta _{2}}),\dots $ be independent identically distributed ${\mathbb{N}^{2}}$-valued random vectors with arbitrarily dependent components. The sequence ${({\Theta _{k}})_{k\in \mathbb{N}}}$ defined by ${\Theta _{k}}={\Pi _{k-1}}\cdot {\eta _{k}}$, where ${\Pi _{0}}=1$ and ${\Pi _{k}}={\xi _{1}}\cdot \dots \cdot {\xi _{k}}$ for $k\in \mathbb{N}$, is called a multiplicative perturbed random walk. Arithmetic properties of the random sets $\{{\Pi _{1}},{\Pi _{2}},\dots ,{\Pi _{k}}\}\subset \mathbb{N}$ and $\{{\Theta _{1}},{\Theta _{2}},\dots ,{\Theta _{k}}\}\subset \mathbb{N}$, $k\in \mathbb{N}$, are studied. In particular, distributional limit theorems for their prime counts and for the least common multiple are derived.

Dalam menyelesaikan soal cerita matematika tidak hanya dibutuhkan jawaban yang benar, tetapi siswa mampu memahami proses dalam memperoleh jawaban soal cerita. Agar guru dapat menilai kemampuan setiap siswa, siswa diharapkan dapat menyelesaikan soal cerita menggunakan tahapan polya secara berurutan. Tahapan tersebut meliputi memahami masalah, menyusun rencana untuk menyelesaikan masalah, melaksanakan rencana yang telah disusun, dan memeriksa ulang jawaban yang diperoleh. Tujuan penelitian ini yaitu untuk menganalisis kemampuan siswa Kelas IV di SDN 152 Gresik dalam menyelesaikan soal cerita matematika pada materi FPB dan KPK. Penelitian ini menggunakan metode penelitian deskriptif kualitatif. Sumber data penelitian ini yaitu siswa Kelas IV SDN 152 Gresik sebanyak 18 Orang. Teknik pengumpulan data yang digunakan peneliti yaitu tes tulis dan wawancara. Teknik analisis data dalam penelitian ini dilakukan melalui 3 tahap meliputi tahap reduksi data, penyajian data, dan penarikan kesimpulan. Berdasarkan hasil analisis data dapat diketahui bahwa kemampuan siswa Kelas IV dalam menyelesaikan soal cerita matematika di SDN 152 Gresik berbeda-beda. Terdapat 1 siswa dengan kemampuan penyelesaian soal cerita tinggi, 9 siswa dengan kemampuan penyelesaian soal cerita sedang, dan 8 siswa dengan kemampuan penyelesaian soal cerita rendah. Siswa dengan kemampuan penyelesaian soal cerita tinggi mampu menyelesaikan semua soal cerita sesuai tahapan polya dengan benar, siswa dengan kemampuan penyelesaian soal cerita sedang hanya mampu menyelesaikan 3-4 soal sesuai tahapan polya dengan benar, dan siswa dengan kemampuan penyelesaian soal cerita rendah hampir mampu menyelesaikan 3 soal cerita sesuai tahapan polya.

We unify in a large class of additive functions the results obtained in the first part of this work. The proof rests on series involving the Riemann zeta function and certain sums of primes which may have their own interest. [ABSTRACT FROM AUTHOR]

Private set union (PSU) allows several parties to obtain the union of their private sets without disclosing each party's private information. Existing PSU protocols often have polynomial complexity for the complete set size or complicated process. In this paper, a quantum multi-party PSU protocol based on least common multiple (LCM) and Shor's algorithm is proposed, which enables the union of multiple sets to be computed all at once. In order to increase the one-time success probability of the protocol, we first improved Shor's period-finding algorithm, which is used in LCM computation and integer factoring. Each party's private set is encoded into an integer obtained by multiplying several prime numbers, thus the PSU problem is transformed into an LCM problem. The LCM of these integers is computed by using the improved Shor's period-finding algorithm, and then factored to derived the union set. We prove the correctness of the proposed protocol, and its unconditional security against semi-honest attacks. Complexity analysis shows that our protocol has logarithmic complexity for the complete set size. [ABSTRACT FROM AUTHOR]

For a wide class of sequences of integer domains D n ⊂ N d , n ∈ N , we prove distributional limit theorems for F (X 1 (n) , … , X d (n)) , where F is a multivariate multiplicative function and (X 1 (n) , … , X d (n)) is a random vector with uniform distribution on D n . As a corollary, we obtain limit theorems for the greatest common divisor and least common multiple of the random set { X 1 (n) , … , X d (n) } . This generalizes previously known limit results for D n being either a discrete cube or a discrete hyperbolic region. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be a fixed integer. We consider sums of type ∑ n 1 ⋯ n k ≤ x F (n 1 , ... , n k) , taken over the hyperbolic region { (n 1 , ... , n k) ∈ N k : n 1 ⋯ n k ≤ x } , where F : N k → C is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations ∑ n 1 ⋯ n k ≤ x f ((n 1 , ... , n k)) and ∑ n 1 ⋯ n k ≤ x f ([ n 1 , ... , n k ]) , involving the GCD and LCM of the integers n 1 , ... , n k , where f : N → C belongs to certain classes of functions. Some of our results generalize those obtained by the authors (Heyman and Tóth in Results Math 76(1): 22, 2021) for k = 2 . [ABSTRACT FROM AUTHOR]

The author's last two papers concerned the study of arithmetic functions related to the greatest common divisor. In the present paper, similarly to the two previous papers we will define new multiplicative arithmetic functionsrelated to the least common multiple and we will study several interesting properties of them. Also, we will discuss the values of two special functionsof them at perfect numbers. [ABSTRACT FROM AUTHOR]

We study the distribution of the least common multiple of positive integers in NΠ[1, x] and related problems. We refine some results of Hilberdink and Tóth (2016). We also give a partial result toward a conjecture of Hilberdink, Luca, and Tóth (2020). [ABSTRACT FROM AUTHOR]

Let n and k be positive integers with $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$. In 1978, Borwein ['A sum of reciprocals of least common multiples', Canad. Math. Bull. 20 (1978), 117–118] confirmed a conjecture of Erdős by showing that $S_{n,1}\le 1-{1}/{2^{n-1}}$. Hong ['A sharp upper bound for the sum of reciprocals of least common multiples', Acta Math. Hungar. 160 (2020), 360–375] improved Borwein's upper bound to $S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for $S_{n,2}$ and $S_{n,3}$. In this paper, we present a sharp upper bound for $S_{n,4}$ and characterise the sequences $\{a_i\}_{i=1}^n$ for which the upper bound is attained. [ABSTRACT FROM AUTHOR]

Landau's function g (n) [1] is the maximal possible least common multiple of positive integers with sum not exceeding n. Under the additional assumptions that these numbers are the differences of disjoint arithmetic progressions, the maximum g ~ (n) was introduced by Okhotin [2]. We find sharp logarithmic asymptotics for g ~ (n) . [ABSTRACT FROM AUTHOR]

A graph G without isolated vertices is a least common multiple of two graphs H1 and H2 if G is a smallest graph, in terms of number of edges, such that there exists a decomposition of G into edge disjoint copies of H1 and H2. The collection of all least common multiples of H1 and H2 is denoted by LCM(H2;H1) and the size of a least common multiple of H1 and H2 is denoted by lcm(H1;H2). In this paper lcm(P4;Cm ❏ Pn), lcm(P4;Wm ❏ Pn) and lcm(P4;Wm ❏ Cn) are determined where the product is the cartesian product. [ABSTRACT FROM AUTHOR]

Let (F n) be the sequence of Fibonacci numbers and, for each positive integer k, let P k be the set of primes p such that gcd (p - 1 , F p - 1) = k . We prove that the relative density r (P k) of P k exists, and we give a formula for r (P k) in terms of an absolutely convergent series. Furthermore, we give an effective criterion to establish if a given k satisfies r (P k) > 0 , and we provide upper and lower bounds for the counting function of the set of such k's. As an application of our results, we give a new proof of a lower bound for the counting function of the set of integers of the form gcd (n , F n) , for some positive integer n. Our proof is more elementary than the previous one given by Leonetti and Sanna, which relies on a result of Cubre and Rouse. [ABSTRACT FROM AUTHOR]

Cilleruelo conjectured that if f ∈ ℤ [ x ] is an irreducible polynomial of degree d ≥ 2 then, log lcm { f (n) | n < x } ∼ (d − 1) x log x. In this paper, we investigate the analog of prime arguments, namely, lcm { f (p) | p < x } , where p denotes a prime and obtain nontrivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of f (p). [ABSTRACT FROM AUTHOR]

Let B n (m) be a set picked uniformly at random among all m -elements subsets of { 1 , 2 , ... , n }. We provide a pathwise construction of the collection (B n (m)) 1 ⩽ m ⩽ n and prove that the logarithm of the least common multiple of the integers in (B n (⌊ m t ⌋)) t ⩾ 0 , properly centered and normalized, converges to a Brownian motion when both m , n tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of m independent random variables having uniform distribution on { 1 , 2 , ... , n }. Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions. [ABSTRACT FROM AUTHOR]

We obtain asymptotic formulas for the sums ∑ n 1 , ... , n k ⩽ x f((n1,... , nk)) and ∑ n 1 , ... , n k ⩽ x f([n1,... , nk]), involving the GCD and LCM of the integers n1,... , nk, where f belongs to certain classes of additive arithmetic functions. In particular, we consider the generalized omega function Ωℓ(n) = ∑ p ν ‖ n v ℓ investigated by Duncan (1962) and Hassani (2018), and the functions A(n) = ∑ p ν ‖ n vp , A∗(n) = ∑p ∣ np, B(n) = A(n) − A∗(n) studied by Alladi and Erdős (1977). As a key auxiliary result, we use an inclusion–exclusion-type identity. [ABSTRACT FROM AUTHOR]

Let $k$ be a positive integer and $f(x)$ a polynomial with integer coefficients. Associated to the least common multiple ${\rm lcm}_{0\le i\le k}\{f(n+i)\}$, we define the function $\mathcal{G}_{k, f}$ for all positive integers $n\in \mathbb{N}^*\setminus Z_{k, f}$ by $\mathcal{G}_{k, f}(n):=\frac{\prod_{i=0}^k |f(n+i)|}{{\rm lcm}_{0\le i\le k}\{f(n+i)\}},$ where $Z_{k,f}:=\bigcup_{i=0}^k\{n\in \mathbb{N}^*: f(n+i)=0\}.$ If $f(x)=x$, then Farhi showed in 2007 that $\mathcal{G}_{k, f}$ is periodic with $k!$ as its period. Consequently, Hong and Yang improved Farhi's period $k!$ to ${\rm lcm}(1,...,k)$. Later on, Farhi and Kane confirmed a conjecture of Hong and Yang and determined the smallest period of $\mathcal{G}_{k, f}$. For the general linear polynomial $f(x)$, Hong and Qian showed in 2011 that $\mathcal{G}_{k, f}$ is periodic and got a formula for its smallest period. In 2015, Hong and Qian characterized the quadratic polynomial $f(x)$ such that $\mathcal{G}_{k, f}$ is almost periodic and also arrived at an explicit formula for the smallest period of $\mathcal{G}_{k, f}$. If $\deg f(x)\ge 3$, then one naturally asks the following interesting question: Is the arithmetic function $\mathcal{G}_{k,f}$ almost periodic and, if so, what is the smallest period? In this paper, we asnwer this question for the case $f(x)=x^3+2$. First of all, with the help of Hua's identity, we prove that $\mathcal{G}_{k,x^3+2}$ is periodic. Consequently, we use Hensel's lemma, develop a detailed $p$-adic analysis to $\mathcal{G}_{k, x^3+2}$ and particularly investigate arithmetic properties of the congruences $x^3+2\equiv 0 \pmod{p^e}$ and $x^6+108\equiv 0\pmod{p^e}$, and with more efforts, its smallest period is finally determined. Furthermore, an asymptotic formula for ${\rm log \ lcm}_{0 \le i \le k}\{(n+i)^3+2\}$ is given.

Let (F n) n ≥ 1 be the sequence of Fibonacci numbers. Guy and Matiyasevich proved that log lcm (F 1 , F 2 , ... , F n) ∼ 3 log α π 2 ⋅ n 2 , as n → + ∞ , where lcm is the least common multiple and α : = 1 + 5) / 2 is the golden ratio. We prove that for every periodic sequence s = (s n) n ≥ 1 in { − 1 , + 1 } there exists an effectively computable rational number C s > 0 such that log lcm (F 3 + s 3 , F 4 + s 4 , ... , F n + s n) ∼ 3 log α π 2 ⋅ C s ⋅ n 2 , as n → + ∞. Moreover, we show that if (s n) n ≥ 1 is a sequence of independent uniformly distributed random variables in { − 1 , + 1 } then [ log lcm (F 3 + s 3 , F 4 + s 4 , ... , F n + s n) ] ∼ 3 log α π 2 ⋅ 1 5 Li 2 (1 / 1 6) 2 ⋅ n 2 , as n → + ∞ , where Li 2 is the dilogarithm function. [ABSTRACT FROM AUTHOR]

A conjecture due to Cilleruelo states that for an irreducible polynomial f with integer coefficients of degree d = 2, the least common multiple Lf (N) of the sequence f(1), f(2), . . ., f(N) has asymptotic growth log Lf (N) ~ (d - 1)N logN as N ?8. We establish a version of this conjecture for almost all shifts of a fixed polynomial, the range of N depending on the range of shifts. [ABSTRACT FROM AUTHOR]

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers a and b , with b ≥ 2 , we have lcm (a , a + b , ... , a + n b) ≤ (c 1 ⋅ b log b) n + ⌊ a b ⌋ (∀ n ≥ b + 1) , where c 1 = 4 1. 3 0 1 4 2. If in addition b is a prime number and a < b , then we prove that for any n ≥ b + 1 , we have lcm (a , a + b , ... , a + n b) ≤ (c 2 ⋅ b b / (b − 1)) n , where c 2 = 1 2. 3 0 6 4 1. Finally, we apply those inequalities to estimate the arithmetic function M defined by M (n) : = 1 φ (n) ∑ 1 ≤ ℓ ≤ n ℓ ∧ n = 1 1 ℓ (∀ n ≥ 1), as well as some values of the generalized Chebyshev function (x ; k , ℓ). [ABSTRACT FROM AUTHOR]

In this paper we investigate the distribution of degrees of the least common multiples of random subsets of monic polynomials of degree n in F q [ t ] . We consider two different probabilistic models and find the concentration of degrees around the corresponding expectations when q → ∞ or n → ∞ . [ABSTRACT FROM AUTHOR]

This paper deals with the number theoretic properties of non-unit elements of the ring Zn. Let D be the set of all non-trivial divisors of a positive integer n. Let D1 and D2 be the subsets of D having the least common multiple which are incongruent to zero modulo n with every other element of D and congruent to zero modulo n with at least one another element of D, respectively. Then D can be written as the disjoint union of D1 and D2 in Zn. We explore the results on these sets based on all the characterizations of n. We obtain a formula for enumerating the cardinality of the set of all non-unit elements in Zn whose principal ideals are equal. Further, we present an algorithm for enumerating these sets of all non-unit elements.

We obtain asymptotic formulas with remainder terms for the hyperbolic summations ∑ m n ≤ x f ((m , n)) and ∑ m n ≤ x f ([ m , n ]) , where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions f (n) = τ (n) , log n , ω (n) and Ω (n) . We also define a common generalization of the latter three functions, and prove a corresponding result. [ABSTRACT FROM AUTHOR]

For every positive integer n and for every α ∈ [ 0 , 1 ] , let B (n , α) denote the probabilistic model in which a random set A ⊆ { 1 , ... , n } is constructed by picking independently each element of { 1 , ... , n } with probability α . Cilleruelo, Rué, Šarka, and Zumalacárregui proved an almost sure asymptotic formula for the logarithm of the least common multiple of the elements of A .Let q be an indeterminate and let [ k ] q : = 1 + q + q 2 + ⋯ + q k - 1 ∈ Z [ q ] be the q-analog of the positive integer k. We determine the expected value and the variance of X : = deg lcm ( [ A ] q) , where [ A ] q : = { [ k ] q : k ∈ A } . Then we prove an almost sure asymptotic formula for X, which is a q-analog of the result of Cilleruelo et al. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. Firstly, to give short proofs of the celebrated results of Rudd (1970), De Marco (1972), Brookshear (1977), Neville (1990), and Vechtomov (1996) with the help of some well-known results in ring theory. Secondly, to establish new algebraic characterizations for when C(X) is von Neumann regular, semihereditary, or a Bézout ring. The notions of the greatest common divisor and least common multiple in the ring C(X) are studied. Finally, the space X, for which every idempotent of the classical quotient ring of C(X) is actually an element of C(X) itself, is completely determined. [ABSTRACT FROM AUTHOR]

Let n and k be positive integers such that n ≥ k + 1 and let { a i } i = 1 n be an arbitrary given strictly increasing sequence of positive integers. Let S n , k : = ∑ i = 1 n - k 1 1 cm (a i , a i + k) . Borwein [3] proved a conjecture of Erdős stating that if n ≥ 2 , then S n , 1 ≤ 1 - 1 2 n - 1 , with the equality holding if and only if a i = 2 i - 1 for 1 ≤ i ≤ n . In this paper, we first improve Borwein's upper bound by showing that S n , 1 ≤ 1 a 1 (1 - 1 2 n - 1 ) with the equality occurring if and only if a i = 2 i - 1 a 1 for all integers 1 ≤ i ≤ n . Then we use this improved upper bound to show that if n ≥ 3 , then S n , 2 ≤ 7 6 + 1 2 ⌊ n 2 ⌋ (2 3 δ n - 7 3) , with the equality holding if and only if a 1 = 1 , a 2 i = 2 i and a 2 i + 1 = 3 × 2 i - 1 for all integers 1 ≤ i ≤ ⌊ n 2 ⌋ , where δ n : = 0 if n is even, and 1 if n is odd. Furthermore, we show that if n ≥ 7 , then S n , 3 ≤ 17 15 - 37 15 · 1 2 ⌊ n 3 ⌋ + ϵ n 2 ⌈ n 3 ⌉ , with equality occurring if and only if a i = i for all i ∈ { 1 , 2 , 3 } and a 3 i + 1 = 2 i + 1 (1 ≤ i ≤ ⌊ n - 1 3 ⌋) , a 3 i + 2 = 5 × 2 i - 1 (1 ≤ i ≤ ⌊ n - 2 3 ⌋) and a 3 i + 3 = 3 × 2 i (1 ≤ i ≤ ⌊ n 3 ⌋ - 1) , where ϵ n = 0 if 3 ∣ n , 1 if n ≡ 1 (mod 3) and 9 5 if n ≡ 2 (mod 3) . We also present a tight upper bound for S n , 3 if n ∈ { 4 , 5 , 6 } . [ABSTRACT FROM AUTHOR]

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of k positive integers, where k ≥ 2 is fixed. We refine and generalize an asymptotic formula of Bordellès [Mean values of generalized gcd-sum and lcm-sum functions, J. Integer Seq. 10 (2007) Article ID:07.9.2, 13 pp], and extend certain related results of Hilberdink and Tóth [On the average value of the least common multiple of k positive integers, J. Number Theory 169 (2016) 327–341]. We also formulate some conjectures and open problems. [ABSTRACT FROM AUTHOR]

Let L n (k) denote the least common multiple of k independent random integers uniformly chosen in { 1 , 2 , ... , n }. In this article, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as n → ∞ of f (L n (k)) n r k for a wide class of multiplicative arithmetic functions f with polynomial growth r ∈ R. Furthermore, we identify the limit as an infinite product of independent random variables indexed by the set of prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Erdős and Wintner (1939), Fernández and Fernández (2013) and Hilberdink and Tóth (2016). [ABSTRACT FROM AUTHOR]

We present a proof of the fact that for n≥0. This result has a standard proof via an integral, but our proof is purely number-theoretic, requiring little more than inductions based on lists. The almost-pictorial proof is based on manipulations of a variant of Leibniz's harmonic triangle, itself a relative of Pascal's better-known Triangle. [ABSTRACT FROM AUTHOR]

In this note we obtain an asymptotic formula with a power saving error term for the summation function of Euler phi-function evaluated at iterated and generalized least common multiples of four integer variables. [ABSTRACT FROM AUTHOR]

We deduce asymptotic formulas for the sums ∑ n 1 , … , n r ≤ x f ( n 1 ⋯ n r ) and ∑ n 1 , … , n r ≤ x f ( [ n 1 , … , n r ] ) , where r ≥ 2 is a fixed integer, [ n 1 , … , n r ] stands for the least common multiple of the integers n 1 , … , n r and f is one of the divisor functions τ 1 , k ( n ) ( k ≥ 1 ), τ ( e ) ( n ) and τ ⁎ ( n ) . Our formulas refine and generalize a result of Lelechenko (2014). A new generalization of the Busche–Ramanujan identity is also pointed out. [ABSTRACT FROM AUTHOR]

Let Fn and Ln be the nth Fibonacci number and Lucas number, re- spectively. The order of appearance of m in the Fibonacci sequence, denoted by z(m), is the smallest positive integer k such that m divides Fk. The formula for z(LnLn+1Ln+2...Ln+k) has been recently obtained by Marques for 1 ≤ k ≤ 3, and by Marques and Trojovský for k = 4. In this article, we extend the results to the cases k = 5 and k = 6. Our method gives a general idea on how to obtain the formulas of z(LnLn+1...Ln+k) for every k ≥ 1. [ABSTRACT FROM AUTHOR]

In this paper we present an identity involving the greatest common divisors of almost all possible subproducts of n nonzero integers. Then we prove this identity, with the help of the fundamental theorem of arithmetic, and an identity concerning the minimum function min. As a consequence, a new formula for the least common multiple is derived. [ABSTRACT FROM AUTHOR]

In this paper we obtain an asymptotic formula with a power saving error term for the summation function of a family of generalized least common multiple and greatest common divisor functions of several integer variables. [ABSTRACT FROM AUTHOR]