Jacobsthal function

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The ordinary Jacobsthal function $j(n)$ is defined as the smallest positive integer $m$, such that every sequence of $m$ consecutive integers contains an integer coprime to $n$. The definition refers to all integers, not just those in the range $\scriptstyle [1,\,n-1]\,$.

A048669 Jacobsthal function: maximal gap in a list of all the integers relatively prime to $n$.

{1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 3, 2, 2, 4, 2, 4, 3, 4, 2, 4, 2, 4, 2, 4, 2, 6, 2, 2, 3, 4, 3, 4, 2, 4, 3, 4, 2, 6, 2, 4, 3, 4, 2, 4, 2, 4, 3, 4, 2, 4, 3, 4, 3, 4, 2, 6, 2, 4, 3, 2, 3, 6, 2, 4, 3, 6, 2, 4, 2, 4, 3, 4, 3, ...}

Asymptotic behavior

Iwaniec proved that

$j(n)=O(\log ^{2}(n)).\,$

Jacobsthal conjectured that

$j(n)=O\left(\left({\frac {\log n}{\log \log n}}\right)^{2}\right).\,$

Jacobsthal function of primorial numbers

The Jacobsthal function of primorial numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h(n)} is defined as the smallest positive integer $m$, such that every sequence of $m$ consecutive integers contains an integer coprime to the product of the first $n$ primes. The definition refers to all integers, not just those in the range $\scriptstyle [1,\,n-1]\,$.

$h(n):=j\left((p_{n})\#\right)=j\left(\prod _{i=1}^{n}p_{i}\right),\,$

where $\scriptstyle (p_{n})\#\,$ is the $n$th primorial number (the product of the first $n$ primes).

A048670 Jacobsthal function A048669 applied to the product of the first $n$ primes (A002110).

{2, 4, 6, 10, 14, 22, 26, 34, 40, 46, 58, 66, 74, 90, 100, 106, 118, 132, 152, 174, 190, 200, 216, 234, 258, 264, 282, 300, 312, 330, 354, 378, 388, 414, 432, 450, 476, 492, 510, ...}

External links

• Basic C-version of Jacobsthal calculation, GitHub—Social Coding.
• THOMAS R. HAGEDORN, COMPUTATION OF JACOBSTHAL'S FUNCTION h(n) FOR n < 50.