A301316 a(n) = ((n-1)! + 1) mod n^2.

0, 2, 3, 7, 0, 13, 35, 49, 64, 81, 11, 1, 0, 57, 1, 1, 85, 1, 38, 1, 1, 133, 184, 1, 1, 521, 1, 1, 522, 1, 589, 1, 1, 885, 1, 1, 259, 381, 1, 1, 656, 1, 559, 1, 1, 553, 282, 1, 1, 1, 1, 1, 1802, 1, 1, 1, 1, 2553, 1593, 1, 3416, 993, 1, 1, 1, 1, 804

Offset: 1

Author

Stanislav Sykora, Mar 18 2018

Keywords

• nonn

Comments

By definition, when n > 1, a(n) = 0 then n is a Wilson prime (A007540).

For a(n) to equal 1, (n-1)! must be divisible by n^2 which is the prevailing case for large n. For example, all n which are a product of more than two distinct primes belong to this category. So do all proper powers of primes except 2^2, 2^3, and 3^2. Obviously, when a(n) = 1, then also A055976(n) = 1.

The cases of a(n) > 1 include, for example, all primes other than Wilson's and all numbers of the form n=2*p, where p is a prime.

Examples

			From _Muniru A Asiru_, Mar 20 2018: (Start)
((1-1)! + 1) mod 1^2 = (0! +1) mod 1 = 2 mod 1 = 0.
((2-1)! + 1) mod 2^2 = (1! +1) mod 4 = 2 mod 4 = 2.
((3-1)! + 1) mod 3^2 = (2! +1) mod 9 = 3 mod 9 = 3.
((4-1)! + 1) mod 4^2 = (3! +1) mod 16 = 7 mod 16 = 7.
((5-1)! + 1) mod 5^2 = (4! +1) mod 25 = 25 mod 25 = 0.
... (End)
		

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..10000
• Wikipedia, Wilson prime

Crossrefs

Cf. A000290, A038507, A007540, A055976, A301317.

Programs

GAP

List([1..60],n->(Factorial(n-1)+1) mod n^2); # Muniru A Asiru, Mar 20 2018

Maple

seq((factorial(n-1)+1) mod n^2,n=1..60); # Muniru A Asiru, Mar 20 2018

Mathematica

Array[Mod[(# - 1)! + 1, #^2] &, 67] (* Michael De Vlieger, Apr 21 2018 *)

PARI

a(n) = ((n-1)! + 1) % n^2; \\ Michel Marcus, Mar 18 2018

Formula

a(n) = ((n-1)! + 1) mod n^2. - Jon E. Schoenfield, Mar 18 2018

a(n) = A038507(n-1) mod A000290(n). - Michel Marcus, Mar 20 2018