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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 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 nth 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, ...}

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