%I #23 Oct 28 2021 12:37:08
%S 2,2,3,23,1669,1895359,111113196467011
%N Smallest prime p such that there is a gap of exactly n! between p and the next prime.
%C a(7) > 1.5 * 10^18. - _Charles R Greathouse IV_, Jun 30 2011
%C a(7) <=
%C 5321252506668526413269161812412779312234715413010708809313699883082142158368298199 (see the Nicely page). - _Abhiram R Devesh_, Aug 09 2014
%H Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101]
%F a(n) = A000230(n!/2) for n > 1. - _Charles R Greathouse IV_, Jun 30 2011
%e a(4) = 1669 because the next prime after 1669 is 1693 and 1693 - 1669 = 24 = 4!
%p with(numtheory):for n from 0 to 10 do:id:=0:for k from 1 to 2000000 while(id=0) do:p1:=ithprime(k):p2:=ithprime(k+1):if p2-p1 = n! then id:=1: printf(`%d, `,p1): else fi:od:od:
%t f[n_] := Block[{k = 1}, While[Prime[k + 1] != n + Prime[k], k++ ]; Prime[k]]; Do[ Print[ f[n!]], {n, 0, 10}]
%o (PARI) a(n)=my(p=2);n=n!;forprime(q=3,default(primelimit),if(q-p==n,return(p));p=q) \\ _Charles R Greathouse IV_, Jun 30 2011
%o (Python)
%o import sympy
%o n=0
%o while n>=0:
%o ....p=2
%o ....while sympy.nextprime(p)-p!=(sympy.factorial(n)):
%o ........p=sympy.nextprime(p)
%o ....print(p)
%o ....n=n+1
%o ....p=sympy.nextprime(p)
%o ## _Abhiram R Devesh_, Aug 09 2014
%K nonn,hard
%O 0,1
%A _Michel Lagneau_, Jun 30 2011
%E a(6) from _Charles R Greathouse IV_, Jun 30 2011