Sieving by prime powers
P. Gallagher (1974)
Acta Arithmetica
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P. Gallagher (1974)
Acta Arithmetica
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Jean-Marie De Koninck, Jason Pierre Sweeney (2001)
Colloquium Mathematicae
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The main objective of this paper is to analyze the unimodal character of the frequency function of the largest prime factor. To do that, let P(n) stand for the largest prime factor of n. Then define f(x,p): = #{n ≤ x | P(n) = p}. If f(x,p) is considered as a function of p, for 2 ≤ p ≤ x, the primes in the interval [2,x] belong to three intervals I₁(x) = [2,v(x)], I₂(x) = ]v(x),w(x)[ and I₃(x) = [w(x),x], with v(x) < w(x), such that f(x,p) increases for p ∈ I₁(x), reaches its maximum...
Nyman, Bertil, Nicely, Thomas R. (2003)
Journal of Integer Sequences [electronic only]
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Yong-Gao Chen (2012)
Acta Arithmetica
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K. Ramachandra (1971)
Acta Arithmetica
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Magdalena Jastrzebska, Adam Grabowski (2006)
Formalized Mathematics
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We formalized some basic properties of the Möbius function which is defined classically as [...] as e.g., its multiplicativity. To enable smooth reasoning about the sum of this number-theoretic function, we introduced an underlying many-sorted set indexed by the set of natural numbers. Its elements are just values of the Möbius function.The second part of the paper is devoted to the notion of the radical of number, i.e. the product of its all prime factors.The formalization (which is...
K. Szymiczek (1964)
Colloquium Mathematicae
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Jiahai Kan (2004)
Acta Arithmetica
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Yingchun Cai (2002)
Acta Arithmetica
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Marco Riccardi (2006)
Formalized Mathematics
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The first four sections of this article include some auxiliary theorems related to number and finite sequence of numbers, in particular a primality test, the Pocklington's theorem (see [19]). The last section presents the formalization of Bertrand's postulate closely following the book [1], pp. 7-9.
Müller, Tom (2005)
Journal of Integer Sequences [electronic only]
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Florian Luca, Francesco Pappalardi (2007)
Acta Arithmetica
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Paul Erdös, Aleksandar Ivić (1982)
Publications de l'Institut Mathématique
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Artūras Dubickas, Andrius Stankevičius (2007)
Acta Arithmetica
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