A061007 a(n) = -(n-1)! mod n.

0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Author

Henry Bottomley, Apr 12 2001

Keywords

• nonn
• easy

Comments

The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002

In particular, this is identical to the isprime function A010051 except for a(4) = 2 instead of 0. This is equivalent to Wilson's theorem, (n-1)! == -1 (mod n) iff n is prime. If n = p*q with p, q > 1, then p, q < n-1 and (n-1)! will contain the two factors p and q, unless p = q = 2 (if p = q > 2 then also 2p < n-1, so there are indeed two factors p in (n-1)!), whence (n-1)! == 0 (mod n). - M. F. Hasler, Jul 19 2024

Examples

			a(4) = 2 since -(4 - 1)! = -6 = 2 mod 4.
a(5) = 1 since -(5 - 1)! = -24 = 1 mod 5.
a(6) = 0 since -(6 - 1)! = -120 = 0 mod 6.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Positive for all but the first term of A046022.

Cf. A000040 (the primes), A000142, A010051 (isprime function), A055976, A061006, A061008, A061009.

Programs

Mathematica

Table[Mod[-(n - 1)!, n], {n, 100}] (* Alonso del Arte, Mar 20 2014 *)

PARI

A061007(n) = ((-((n-1)!))%n); \\ Antti Karttunen, Aug 27 2017

PARI

apply( {A061007(n) = !(n-1)!%n}, [0..99]) \\ M. F. Hasler, Jul 19 2024

Python

from sympy import isprime
def A061007(n): return 2 if n == 4 else int(isprime(n)) # Chai Wah Wu, Mar 22 2023

Formula

a(4) = 2, a(p) = 1 for p prime, a(n) = 0 otherwise. Apart from n = 4, a(n) = A010051(n) = A061006(n)/(n-1).