This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The exact subscript of the 5th prime [1 + 27! = 10888869450418352160768000001] is not yet available.
10 is in the sequence because 10! = 3628800, NextPrime(10) = 11 and 3628800 - 11 = 3628789 is prime.
Do[If[PrimeQ[k!-NextPrime[k]], Print[k]], {k, 0, 1425}] (* Farideh Firoozbakht, Feb 26 2004 *)
[n: n in [5..500] | IsPrime((Factorial(n)-5) div 5)]; // Vincenzo Librandi, Nov 21 2016
a = {}; Do[If[PrimeQ[(n! - 5)/5], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a (* Artur Jasinski *)
a:=proc(n) if isprime((1/6)*factorial(n)-1)=true then n else end if end proc: seq(a(n),n=4..500); # Emeric Deutsch, Apr 29 2008
a = {}; Do[If[PrimeQ[(n! - 6)/6], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a (* Artur Jasinski *)
a = {}; Do[If[PrimeQ[(n! - 7)/7], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a (*Artur Jasinski*)
a:=proc(n) if isprime((1/8)*factorial(n)-1)=true then n else end if end proc: seq(a(n),n=4..550); # Emeric Deutsch, May 07 2008
a = {}; Do[If[PrimeQ[(n! - 8)/8], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a
a = {}; Do[If[PrimeQ[(n! - 9)/9], Print[a]; AppendTo[a, n]], {n, 1, 300}]; a
for(n=1,1000,if(floor(n!/9-1)==n!/9-1,if(ispseudoprime(n!/9-1),print(n)))) \\ Derek Orr, Mar 28 2014
4$ - 1 = 6 - 1 = 5 is prime, so 4 is in the sequence.
a := proc(n) select(x -> isprime(A056040(x)-1),[$0..n]) end:
fQ[n_] := PrimeQ[ -1 + 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]]; Select[ Range@ 3647, fQ] (* Robert G. Wilson v, Aug 09 2010 *)
is(k) = ispseudoprime(k!/(k\2)!^2-1); \\ Jinyuan Wang, Mar 22 2020
The a(4) = 4 and a(5) = 18 permutations:
(1,1,1,2) (1,1,1,2,3)
(1,1,2,1) (1,1,1,3,2)
(1,2,1,1) (1,1,2,1,3)
(2,1,1,1) (1,1,2,3,1)
(1,1,3,1,2)
(1,1,3,2,1)
(1,2,1,1,3)
(1,2,3,1,1)
(1,3,1,1,2)
(1,3,2,1,1)
(2,1,1,1,3)
(2,1,1,3,1)
(2,1,3,1,1)
(2,3,1,1,1)
(3,1,1,1,2)
(3,1,1,2,1)
(3,1,2,1,1)
(3,2,1,1,1)
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n!]],MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}]
\\ See A335452 for count. a(n)={my(sig=factor(n!)[, 2]); vecsum(sig)!/vecprod([k! | k<-sig]) - count(sig)} \\ Andrew Howroyd, Apr 17 2021
a(10) = 6 because 10!/6! - 1 = 5039 is prime, whereas 10!/k! - 1 is composite for k = 1..5.
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