1,721,064 research outputs found
On the asymptotic density of the support of a Dirichlet convolution
Full text linkLet ν be a multiplicative arithmetic function with support of positive asymptotic density. We prove that for any not identically zero arithmetic function f such that ∑f(n) ≠01/n < ∞ , the support of the Dirichlet convolution f * ν possesses a positive asymptotic density. When f is a multiplicative function, we give also a quantitative version of this claim. This generalizes a previous result of P. Pollack and the author, concerning the support of Möbius and Dirichlet transforms of arithmetic functions. © 2013 Elsevier Inc
On the l.c.m. of random terms of binary recurrence sequences
Full text linkFor every positive integer n and every δ∈[0,1], let B(n,δ) denote the probabilistic model in which a random set A⊆1,...,n is constructed by choosing independently every element of 1,...,n with probability δ. Moreover, let (uk)k≥0 be an integer sequence satisfying uk=a1uk−1+a2uk−2, for every integer k≥2, where u0=0, u1≠0, and a1,a2 are fixed nonzero integers; and let α and β, with |α|≥|β|, be the two roots of the polynomial X2−a1X−a2. Also, assume that α/β is not a root of unity. We prove that, as δn/logn→+∞, for every A in B(n,δ) we have loglcm(ua:a∈A)∼[Formula presented]⋅n2 with probability 1−o(1), where lcm denotes the lowest common multiple, Li2 is the dilogarithm, and the factor involving δ is meant to be equal to 1 when δ=1. This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Mátyás, who studied the deterministic case δ=1, and is motivated by an asymptotic formula for lcm(A) due to Cilleruelo, Rué, Šarka, and Zumalacárregui
Uncertainty principles connected with the Mobius inversion formula
Full text linkWe say that two arithmetic functions and form a \emph{M\"{o}bius pair} if for all natural numbers . In that case, can be expressed in terms of by the familiar M\"{o}bius inversion formula of elementary number theory. In a previous paper, the first-named author showed that if the members and of a M\"{o}bius pair are both finitely supported, then both functions vanish identically. Here we prove two significantly stronger versions of this uncertainty principle. A corollary of our results is that in a nonzero M\"{o}bius pair, one cannot have both $\sum_{f(n) \neq 0}\frac{1}{n
On arithmetic progressions of integers with a distinct sum of digits
No full textLet b ≥ 2 be a fixed integer. Let sb(n) denote the sum of digits of the nonnegative integer n in the base-b representation. Further let q be a positive integer. In this paper we study the length k of arithmetic progressions n, n + q,..., n + q(k - 1) such that sb(n), sb(n + q),..., sb(n + q(k - 1)) are (pairwise) distinct. More specifically, let Lb,q denote the supremum of k as n varies in the set of nonnegative integers N. We show that Lb,q is bounded from above and hence finite. Then it makes sense to define μb,q as the smallest n ∈ N such that one can take k = Lb,q. We provide upper and lower bounds for μb,q. Furthermore, we derive explicit formulas for Lb,1 and μb,1. Lastly, we give a constructive proof that Lb,q is unbounded with respect to q
Covering an arithmetic progression with geometric progressions and vice versa
Full text linkWe show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression A of real numbers and a sufficiently large integer n (depending on A), there is a need of at least Cn geometric progressions to cover the first n terms of A. A similar result is presented, with the role of arithmetic and geometric progressions reversed. © 2014 World Scientific Publishing Company
Practical numbers in Lucas sequences
Full text linkA practical number is a positive integer n such that all the positive integers m ≤ n can be written as a sum of distinct divisors of n. Let (un)n≥0 be the Lucas sequence satisfying u0 = 0, u1 = 1, and un+2 = aun+1 + bun for all integers n ≥ 0, where a and b are fixed nonzero integers. Assume a(b + 1) even and a2 + 4b > 0. Also, let (Figure presented.) be the set of all positive integers n such that |un| is a practical number. Melfi proved that (Figure presented.) is infinite. We improve this result by showing that # (Figure presented.) (x) ≫ x/log x for all x ≥ 2, where the implied constant depends on a and b. We also pose some open questions regarding (Figure presented.)
On numbers n relatively prime to the nth term of a linear recurrence
Full text linkLet (un)n≥0 be a nondegenerate linear recurrence of integers, and let A be the set of positive integers n such that u n and n are relatively prime. We prove that A has an asymptotic density, and that this density is positive unless (un/n)n≥1 is a linear recurrence
On the greatest common divisor of n and nth Fibonacci number
No full textLet A be the set of all integers of the form gcd(n; Fn), where n is a positive integer and Fn denotes the nth Fibonacci number. We prove that #(A ∩ [1; x]) ≫ x= log x for all x ≥ 2 and that A has zero asymptotic density. Our proofs rely upon a recent result of Cubre and Rouse [5] which gives, for each positive integer n, an explicit formula for the density of primes p such that n divides the rank of appearance of p, that is, the smallest positive integer k such that p divides Fk
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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