Displaying similar documents to “Automaticity IV : sequences, sets, and diversity”

On the powerful part of n 2 + 1

Jan-Christoph Puchta (2003)

Archivum Mathematicum

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We show that n 2 + 1 is powerfull for O ( x 2 / 5 + ϵ ) integers n ≤ x at most, thus answering a question of P. Ribenboim.

Note on special arithmetic and geometric means

Horst Alzer (1994)

Commentationes Mathematicae Universitatis Carolinae

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We prove: If A ( n ) and G ( n ) denote the arithmetic and geometric means of the first n positive integers, then the sequence n ↦ n A ( n ) / G ( n ) - ( n - 1 ) A ( n - 1 ) / G ( n - 1 ) ( n ≥ 2 ) is strictly increasing and converges to e / 2 , as n tends to ∞ .

Composite positive integers whose sum of prime factors is prime

Florian Luca, Damon Moodley (2020)

Archivum Mathematicum

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In this note, we show that the counting function of the number of composite positive integers n ≤ x such that β ( n ) = ∑ p ∣ n p is a prime is of order of magnitude at least x / ( log x ) 3 and at most x / log x .

New bounds on the length of finite pierce and Engel series

P. Erdös, J. O. Shallit (1991)

Journal de théorie des nombres de Bordeaux

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Every real number x , 0 < x ≤ 1 , has an essentially unique expansion as a Pierce series : x = 1 x 1 - 1 x 1 x 2 + 1 x 1 x 2 x 3 - ⋯ where the x i form a strictly increasing sequence of positive integers. The expansion terminates if and only if x is rational. Similarly, every positive real number y has a unique expansion as an Engel series : y = 1 y 1 - 1 y 1 y 2 + 1 y 1 y 2 y 3 + ⋯ where the y i form a (not necessarily strictly) increasing sequence of positive integers. If the expansion is infinite, we require that the sequence yi...

Covers in p -adic analytic geometry and log covers I: Cospecialization of the ( p ′ ) -tempered fundamental group for a family of curves

Emmanuel Lepage (2013)

Annales de l’institut Fourier

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The tempered fundamental group of a p -adic analytic space classifies covers that are dominated by a topological cover (for the Berkovich topology) of a finite étale cover of the space. Here we construct cospecialization homomorphisms between ( p ′ ) versions of the tempered fundamental groups of the fibers of a smooth family of curves with semistable reduction. To do so, we will translate our problem in terms of cospecialization morphisms of fundamental groups of the log fibers of the log...

Uniform distribution modulo one and binary search trees

Michel Dekking, Peter Van der Wal (2002)

Journal de théorie des nombres de Bordeaux

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Any sequence x = ( x k ) k = 1 ∞ of distinct numbers from [0,1] generates a binary tree by storing the numbers consecutively at the nodes according to a left-right algorithm (or equivalently by sorting the numbers according to the Quicksort algorithm). Let H n ( x ) be the height of the tree generated by x 1 , ⋯ , x n . Obviously log n log 2 - 1 ≤ H n ( x ) ≤ n - 1 . If the sequences x are generated by independent random variables having the uniform distribution on [0, 1], then it is well known that there exists c > 0 such that...

On the Piatetski-Shapiro-Vinogradov theorem

Angel Kumchev (1997)

Journal de théorie des nombres de Bordeaux

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In this paper we consider the asymptotic formula for the number of the solutions of the equation p 1 + p 2 + p 3 = N where N is an odd integer and the unknowns p i are prime numbers of the form p i = [ n 1 / γ i ] . We use the two-dimensional van der Corput’s method to prove it under less restrictive conditions than before. In the most interesting case γ 1 = γ 2 = γ 3 = γ our theorem implies that every sufficiently large odd integer N may be written as the sum of three Piatetski-Shapiro primes of type γ for 50 / 53 < γ < 1 . ...