# Jacobsthal function

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The **Jacobsthal function** is $j(n)$ is the smallest integer $k$ such that every sequence of consecutive integers of size $k$ has at least one integer that is relatively prime to $n$ (see https://oeis.org/wiki/Jacobsthal_function)

The Jacobsthal function can also be applied to primorials $p_n\#$ which can be used to the smallest integer $k$ such that every sequence of consecutive integers of size $k$ includes one integer that is $p_{n+1}$-rough. (see https://oeis.org/wiki/Jacobsthal_function)

A list of $k$ for the Jacobsthal function can be found in the Online Encyclopedia of Integer Sequences: (see https://oeis.org/A048670)

Properties:

1. Iwaniec was able to show an upper limit for the Jacobsthal function:

2. Jacobsthal made two conjectures about the the Jacobsthal function. Interestingly, one of those conjectures has been disproven.

3. The Jacobsthal function is challenging to generate. (see MATHEMATICS OF COMPUTATION Volume 78, Number 266, April 2009, Pages 1073–1087 S 0025-5718(08)02166-2 Article electronically published on November 20, 2008)

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## External links

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