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On the consecutive prime divisors of an integer
Functiones et Approximatio Commentarii Mathematici, 2021Paul Erd˝os, Janos Galambos and others have studied the relative size of the consecutive prime divisors of an integer. Here, we further extend this study by examining the distribution of the consecutive neighbour spacings between the prime divisors p 1 (
J. Koninck, Imre Kátai Imre Kátai
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The Ramanujan Journal, 1998
Denote by \( G = {\mathbb Q}^\times/\Gamma \), where \( \Gamma \) is the subgroup of \( {\mathbb Q}^\times \) generated by the shifted primes \( (p+1) \). \( \displaystyle\Gamma = \bigcup_{k=1}^\infty S_k, \) where \( S_k \) is the set of products \( \{(p_1 + 1)^{\varepsilon_1} \cdots (p_k + 1)^{\varepsilon_k}, \varepsilon_i \in \{-1, 0, 1\}\}.
Berrizbeitia, P., Elliott, P. D. T. A.
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Denote by \( G = {\mathbb Q}^\times/\Gamma \), where \( \Gamma \) is the subgroup of \( {\mathbb Q}^\times \) generated by the shifted primes \( (p+1) \). \( \displaystyle\Gamma = \bigcup_{k=1}^\infty S_k, \) where \( S_k \) is the set of products \( \{(p_1 + 1)^{\varepsilon_1} \cdots (p_k + 1)^{\varepsilon_k}, \varepsilon_i \in \{-1, 0, 1\}\}.
Berrizbeitia, P., Elliott, P. D. T. A.
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MULTIPLICATIVE FUNCTIONS ON THE SET OF SHIFTED PRIME NUMBERS
Mathematics of the USSR-Izvestiya, 1992In a recent paper [Acta Arith. 58, No.\ 2, 113-131 (1991; Zbl 0726.11060)], the author investigated the behavior of additive arithmetic function on sequences of ``shifted primes'' \(\{p+a\}\), where \(a\) is a non-zero integer. In the present paper, the author applies similar ideas to obtain a number of results concerning the behavior of multiplicative
N. M. Timofeev
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A Hardy–Ramanujan-type inequality for shifted primes and sifted sets
Lithuanian Mathematical Journal, 2021We establish an analog of the Hardy–Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p + a below x with k distinct prime factors ...
Kevin Ford
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Publicationes Mathematicae Debrecen, 2002
Summary: Let \(p\) range over the set of primes and let \(a\) be a non-zero integer. Here we prove that many properties of the divisors of the natural numbers which can be expressed by inequalities are true for the set \(\{p+a\}\) of shifted primes, too. Among other results we obtain estimates for the quantities \[ \begin{gathered} |\{p:p+a\leq x,\;d_m(
Indlekofer, Karl-Heinz +1 more
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Summary: Let \(p\) range over the set of primes and let \(a\) be a non-zero integer. Here we prove that many properties of the divisors of the natural numbers which can be expressed by inequalities are true for the set \(\{p+a\}\) of shifted primes, too. Among other results we obtain estimates for the quantities \[ \begin{gathered} |\{p:p+a\leq x,\;d_m(
Indlekofer, Karl-Heinz +1 more
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Character sums over shifted primes
Mathematical Notes, 2010Let \(q\) be an arbitrary positive integer, \(\chi\) be a primitive character modulo \(q\), and let \(\Lambda(n)\) be the von Mangoldt function. For \(N\leq q^{16/9}\), the authors prove that \[ \left|\sum_{n\leq N}\Lambda(n)\chi(n+a)\right|\leq \left(N^{7/8}q^{1/9}+N^{33/32}q^{-1/18}\right)q^{o(1)}. \]
Friedlander, J. B. +2 more
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Irrelevant auditory attention shifts prime corresponding responses
Psychological Research, 2003In this paper we investigate whether an attention shift towards an auditory signal, while performing a two-choice serial reaction time task, primes responses in the direction of the auditory signal. In Experiment 1, subjects had to react to the pitch of the signal, which was randomly presented to the left or right ear.
Wim, Notebaert, Eric, Soetens
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Prospects in anode materials for sodium ion batteries - A review
, 2020With the rapid expansion in energy demands and depletion of fossil fuel reservoirs, the importance of clean energy production and storage has increased drastically.
T. Perveen +5 more
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Shifting Lucas Sequences Away from Primes
Summary: We strengthen a result of Jones by showing that for any positive integer \(P\), the Lucas sequence \((U_n)_n\) defined by \(U_0 = 0\), \(U_1 = 1\), \(U_n=P \cdot U_{n -1} + U_{n - 2}\) can be translated by a positive integer \(K(P)\) such that the shifted sequence with general term \(U_n + K(P)\) contains no primes, nor terms one unit away ...Ismailescu, Dan +4 more
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Bounds of Multiplicative Character Sums over Shifted Primes
Proceedings of the Steklov Institute of Mathematics, 2021zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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