A301431 Least nonnegative integer k such that (n!)^2 + n + k + 1 is prime.

0, 0, 0, 1, 6, 1, 4, 3, 4, 13, 6, 1, 46, 9, 16, 7, 24, 41, 48, 9, 10, 81, 366, 35, 82, 21, 100, 39, 152, 71, 66, 377, 4, 27, 8, 25, 10, 225, 70, 13, 158, 125, 294, 3, 86, 81, 26, 133, 208, 141, 50, 31, 26, 127, 112, 173, 802, 363, 374, 47, 910, 437, 74, 213, 1044, 13, 1962, 41, 160, 169, 296, 29

Offset: 0

Author

Seiichi Manyama, Mar 21 2018

Keywords

• nonn

Comments

The (n-1) consecutive numbers (n)^2! + 2, ..., (n!)^2 + n (for n >= 2) are not prime powers (cf. A246655).

Examples

			a(0)=0 because (0!)^2 + 0 + 0 + 1 =   2 is prime.
a(1)=0 because (1!)^2 + 1 + 0 + 1 =   3 is prime.
a(2)=0 because (2!)^2 + 2 + 0 + 1 =   7 is prime.
a(3)=1 because (3!)^2 + 3 + 1 + 1 =  41 is prime and 40 is not prime.
a(4)=6 because (4!)^2 + 4 + 6 + 1 = 587 is prime and 581, 582, ... , 586 are not prime.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A046029, A090786, A246655.

Programs

PARI

a(n) = apply(x->(nextprime(x)-x), (n!)^2+n+1); \\ Altug Alkan, Mar 21 2018