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Relevant bibliographies by topics / Least integer function

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Journal articles

Academic literature on the topic 'Least integer function'

Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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Journal articles on the topic "Least integer function"

1

Hong, Shaofang, and Guoyou Qian. "The least common multiple of consecutive arithmetic progression terms." Proceedings of the Edinburgh Mathematical Society 54, no. 2 (2011): 431–41. http://dx.doi.org/10.1017/s0013091509000431.

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AbstractLet k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n byIf we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to

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2

QIAN, GUOYOU, QIANRONG TAN, and SHAOFANG HONG. "THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN A QUADRATIC PROGRESSION." Bulletin of the Australian Mathematical Society 86, no. 3 (2012): 389–404. http://dx.doi.org/10.1017/s0004972712000202.

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AbstractLet k be any given positive integer. We define the arithmetic function gk for any positive integer n by We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}.

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3

Ma, Wu-Xia, Yong-Gao Chen, and Bing-Ling Wu. "Distribution of the primes involving the ceiling function." International Journal of Number Theory 15, no. 03 (2019): 597–611. http://dx.doi.org/10.1142/s1793042119500313.

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The distribution of the primes of the forms [Formula: see text] and [Formula: see text] are studied extensively, where [Formula: see text] denotes the largest integer not exceeding [Formula: see text]. In this paper, we will consider several new type problems on the distribution of the primes involving the ceiling (floor) function. For any real number [Formula: see text] with [Formula: see text], let [Formula: see text] be the number of integers [Formula: see text] with [Formula: see text] such that [Formula: see text] is prime and let [Formula: see text] be the number of primes [Formula: see

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4

Leonetti, Paolo, and Carlo Sanna. "A note on primes in certain residue classes." International Journal of Number Theory 14, no. 08 (2018): 2219–23. http://dx.doi.org/10.1142/s1793042118501336.

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Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

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5

Ballantine, Cristina, and Mircea Merca. "Combinatorial proof of the minimal excludant theorem." International Journal of Number Theory 17, no. 08 (2021): 1765–79. http://dx.doi.org/10.1142/s1793042121500615.

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The minimal excludant of a partition [Formula: see text], [Formula: see text], is the smallest positive integer that is not a part of [Formula: see text]. For a positive integer [Formula: see text], [Formula: see text] denotes the sum of the minimal excludants of all partitions of [Formula: see text]. Recently, Andrews and Newman obtained a new combinatorial interpretation for [Formula: see text]. They showed, using generating functions, that [Formula: see text] equals the number of partitions of [Formula: see text] into distinct parts using two colors. In this paper, we provide a purely combi

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6

Du, Julia Q. D., Edward Y. S. Liu, and Jack C. D. Zhao. "Congruence properties of pk(n)." International Journal of Number Theory 15, no. 06 (2019): 1267–90. http://dx.doi.org/10.1142/s1793042119500714.

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We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see

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7

Popkov, Kirill Andreevich. "On self-correcting logic circuits of unreliable gates." Keldysh Institute Preprints, no. 49 (2021): 1–18. http://dx.doi.org/10.20948/prepr-2021-49.

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The following statements are proved: 1) for any integer m ≥ 3 there is a basis consisting of Boolean functions of no more than m variables, in which any Boolean function can be implemented by a logic circuit of unreliable gates that self-corrects relative to certain faults in an arbitrary number of gates; 2) for any positive integer k there are bases consisting of Boolean functions of no more than two variables, in each of which any Boolean function can be implemented by a logic circuit of unreliable gates that self-correct relative to certain faults in no more than k gates; 3) there is a func

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8

Lu, Qian, and Qilong Liao. "Normal criterion and shared values by derivatives of meromorphic functions." Tamkang Journal of Mathematics 45, no. 2 (2014): 109–17. http://dx.doi.org/10.5556/j.tkjm.45.2014.1014.

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Let $\mathscr{F}$ be a family of meromorphic functions in a plane domain $D$. If for every function $f\in\mathscr{F}$, all of whose zeros have,at least,multiplicity $l$ and poles have, at least,multiplicity $p$, and for each pair functions $f$ and $g$ in $\mathscr{F}$, $f^{(k)}$ and $g^{(k)}$ share 1 in $D$, where $k,l,$ and $p$ are three positive integer satisfying $\frac{k+1}{l}+\frac{1}{p}\leq 1$, then $\mathscr{F}$ is normal.

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9

Cusick, T. W. "Units in real cyclic quartic fields." Mathematical Proceedings of the Cambridge Philosophical Society 107, no. 1 (1990): 5–17. http://dx.doi.org/10.1017/s0305004100068328.

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Let F be a totally real quartic field. For any α in F, let α, α′, α″, α‴ or α(0) = α(1), α(2), α(3) denote the conjugates of α. Define the function T(α) byWe define a triple of units ε1, ε2, ε3 in F as follows. Let ε1 be a unit which gives the least value of T(ε) for any unit ε ≠ = ± 1 in F. Let ε2 be a unit which gives the least value of T(ε) for any unit ε ≠ = ± ε1m with m a rational integer. Let ε3 be a unit which gives the least value of T(ε) for ε ≠ = ± ε1m ε2n with m and n rational integers. We call ε1, ε2, ε3 the successive unit minima for T(ε).

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10

Chen, Jun-Fan. "Exceptional functions and normal families of holomorphic functions with multiple zeros." gmj 18, no. 1 (2011): 31–38. http://dx.doi.org/10.1515/gmj.2011.0005.

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Abstract Let k be a positive integer, and let ℱ be a family of functions holomorphic on a domain D in C, all of whose zeros are of multiplicity at least k + 1. Let h be a function meromorphic on D, h ≢ 0, ∞. Suppose that for each ƒ ∈ ℱ, ƒ(k)(z) ≠ h(z) for z ∈ D. Then ℱ is a normal family on D. The condition that the zeros of functions in ℱ are of multiplicity at least k + 1 cannot be weakened, and the corresponding result for families of meromorphic functions is no longer true.

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