I noticed that $41$ plus primorials is prime when I add $2$ to $41=43$, then $6$ to $41= 47$, then $30+41=71$ all the way up to $41+9699690=9699731$. Hence I can add the first $8$ primorials to get primes. Do you know if there is a different prime that will allow the addition of more than the first $8$ primorials?
2026-03-28 22:25:47.1774736747
41 +primorial is prime
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I just wrote a little Maple script to find some examples of primorially separated primes, as I will call them. Here are some results:
p=3 is the smallest prime which has a run of 1 primorially separated primes after it.
p=5 is the smallest prime which has a run of 2 primorially separated primes after it.
p=11 is the smallest prime which has a run of 3 primorially separated primes after it.
p=17 is the smallest prime which has a run of 4 primorially separated primes after it.
p=41 is the smallest prime which has a run of 8 primorially separated primes after it.
p=86351 is the smallest prime which has a run of 10 primorially separated primes after it.
Finding a run of 11 or more is turning out to take quite some time, but if my script finds anything I will update further.