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Displaying 161 – 180 of 218

On the largest prime factor of $n! + 2^{n} - 1$

Florian Luca, Igor E. Shparlinski (2005)

Journal de Théorie des Nombres de Bordeaux

For an integer $n \geq 2$ we denote by $P (n)$ the largest prime factor of $n$ . We obtain several upper bounds on the number of solutions of congruences of the form $n! + 2^{n} - 1 \equiv 0 (mod q)$ and use these bounds to show that $\underset{n \to \infty}{lim sup} P (n! + 2^{n} - 1) / n \geq (2 π^{2} + 3) / 18 .$

On the largest prime factors of n and n+1.

Carl Pomerance, Paul Erdös (1978)

Aequationes mathematicae

On the largest prime factors of n and n+1. (Short Communication).

Carl Pomerance, Paul Erdös (1978)

Aequationes mathematicae

On the Laurent Coefficients of Certain Dirichlet Series

Aleksandar Ivić (1993)

Publications de l'Institut Mathématique

On the limiting distribution of a generalized divisor problem for the case -1/2 < a < 0

Yuk-Kam Lau (2001)

Acta Arithmetica

On the limiting distribution of a generalized divisor problem for the case -1 ≤ a < -1/2

Yuk-Kam Lau (2001)

Acta Arithmetica

On the local behaviour of some arithmetical functions

R. Hall, G. Tenenbaum (1984)

Acta Arithmetica

On the local distribution of the iterated divisor function.

Kátai, I., Subbarao, M.V. (2004)

Mathematica Pannonica

On the mean square of the divisor function in short intervals

Aleksandar Ivić (2009)

Journal de Théorie des Nombres de Bordeaux

We provide upper bounds for the mean square integral $\int_{X}^{2 X} {(𝔻_{k} (x + h) - 𝔻_{k} (x))}^{2} d x,$ where $h = h (X) ≫ 1, h = o (x) as X \to \infty$ and $h$ lies in a suitable range. For $k \geq 2$ a fixed integer, $𝔻_{k} (x)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_{k} (n)$ , generated by $ζ^{k} (s)$ .