A340802 Number of composite numbers between the largest noncomposite number <= n! and the smallest noncomposite number >= n!.

0, 0, 1, 5, 13, 7, 11, 53, 29, 21, 13, 29, 89, 19, 89, 75, 89, 77, 189, 59, 61, 103, 185, 203, 189, 95, 43, 167, 253, 107, 187, 79, 37, 289, 173, 257, 97, 43, 169, 135, 131, 175, 179, 155, 291, 189, 311, 155, 141, 157, 449, 119, 129, 349, 131, 609, 383, 391, 429

Offset: 1

Author

Alois P. Heinz, Jan 21 2021

Keywords

• nonn

Examples

			a(4) = 5: 24, 25, 26, 27, 28.
a(5) = 13: 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126.
a(6) = 7: 720, 721, 722, 723, 724, 725, 726.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000040, A002808, A008578, A046933, A054588, A058054, A340013.

Programs

Maple

prevprime(2):= 1:
a:= n-> (f-> max(nextprime(f-1)-prevprime(f+1)-1, 0))(n!):
seq(a(n), n=1..64);

Mathematica

a[n_] := If[n<3, 0, NextPrime[n!] - NextPrime[n!, -1] - 1];
Array[a, 100] (* Jean-François Alcover, Jan 29 2021 *)

Formula

a(n) = A058054(n)-1 for n >= 2.

a(n) = A054588(n)-1 for n >= 3.

a(n) = 2 * A340013 - 1 for n >= 3.