A301427 Least nonnegative integer k such that n! - n - k is prime.

0, 1, 2, 5, 10, 23, 4, 1, 2, 1, 10, 3, 32, 37, 42, 23, 82, 11, 10, 51, 66, 49, 124, 11, 16, 73, 2, 49, 30, 131, 14, 159, 78, 91, 60, 41, 34, 43, 90, 37, 66, 65, 8, 43, 32, 55, 10, 47, 128, 15, 6, 73, 6, 405, 220, 51, 78, 79, 10, 9, 38, 295, 62, 251, 124, 183, 34, 27, 680, 91, 300

Offset: 3

Author

Seiichi Manyama, Mar 21 2018

Keywords

• nonn

Comments

The (n-1) consecutive numbers n!-n, ... , n!-2 (for n > 3) are not prime.

Examples

			a(3)=0 because 3! - 3 - 0 =   3 is prime.
a(4)=1 because 4! - 4 - 1 =  19 is prime and 20 is not.
a(5)=2 because 5! - 5 - 2 = 113 is prime and 114 and 115 are not prime.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 3..500

Crossrefs

Cf. A037155, A073443, A090786.

Programs

Maple

f:= proc(n) local r; r:= n!-n;
  r - prevprime(r)
end proc:
f(3):= 0:
seq(f(i),i=3..100); # Robert Israel, Mar 23 2018

Mathematica

a[n_] := n! - NextPrime[n! - 1, -1] - n;
a /@ Range[3, 100] (* Jean-François Alcover, Oct 26 2020 *)

PARI

a(n) = apply(x->(x-precprime(x)), n!-n);
vector(99, n, a(n+2)) \\ Altug Alkan, Mar 21 2018

Formula

a(n) = A037155(n) - n.