A249355 Remainder of n!+2 divided by n+2.

1, 0, 0, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2

Offset: 0

Author

M. F. Hasler, Oct 26 2014

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 0..2048

Crossrefs

Cf. A055976, A242707.

Programs

Magma

[(Factorial(n)+2) mod(n+2): n in [0..100]]; // Vincenzo Librandi, Oct 27 2014

Mathematica

Table[Mod[n!+2,n+2],{n,0,100}] (* Harvey P. Dale, Sep 01 2022 *)

PARI

a(n)=lift(prod(k=2,n,k,Mod(1,n+2))+2)

PARI

A249355(n)=if(n>2,isprime(n+2)+2,!n) \\ M. F. Hasler, Oct 31 2014

Formula

If n+2 = p > 4 is prime, then a(n) = 3. Indeed, it is known that (p-2)! = 1 (mod p) for all primes p. Thus n!+2 = 1+2 = 3 (mod n+2).

If n+2 is composite and n > 2 then a(n) = 2. There are two cases: n+2 = a*b with a < b <= n (so n! is divisible by a*b), or n+2 = a^2 with 2*a <= n (so n! is divisible by a*(2*a)). - Robert Israel, Oct 27 2014