OEIS A076689

Is defined as smallest integer $a(n)=k$ such that $k n\#+1$ is prime, where $n\#$ is primorial, the product of the first $n$ primes.

Lower bound appears $1$, the primorial primes.

What is upper bound for $a(n)$, possibly using plausible conjectures?

By examining the B-file, it might as as low as $Cn$.

OEIS A076689

Is defined as smallest integer $a(n)=k$ such that $k n\#+1$ is prime, where $n\#$ is primorial, the product of the first $n$ primes.

Lower bound appears $1$, the primorial primes.

What is upper bound for $a(n)$, possibly using plausible conjectures?

By examining the B-file, it might as low as $Cn$.

Upper bound polynomial in $n$ might give fast deterministic algorithm for finding large primes: http://michaelnielsen.org/polymath1/index.php?title=Finding_primes

OEIS A076689

Is defined as smallest integer $k$ such that $a(n)=k n\#+1$ is prime, where $n\#$ is primorial, the product of the first $n$ primes.

Lower bound appears $1$, the primorial primes.

What is upper bound for $a(n)$, possibly using plausible conjectures?

By examining the B-file, it might as as low as $Cn$.

OEIS A076689

Is defined as smallest integer $a(n)=k$ such that $k n\#+1$ is prime, where $n\#$ is primorial, the product of the first $n$ primes.

Lower bound appears $1$, the primorial primes.

What is upper bound for $a(n)$, possibly using plausible conjectures?

By examining the B-file, it might as as low as $Cn$.