login
Remainder of n!+2 divided by n+2
1

%I #29 Sep 08 2022 08:46:10

%S 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,

%T 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,

%U 2,3,2,3,2,2,2,2,2,3,2,2,2,3,2,2,2,2,2,3,2,2

%N Remainder of n!+2 divided by n+2

%H Antti Karttunen, <a href="/A249355/b249355.txt">Table of n, a(n) for n = 0..2048</a>

%F 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).

%F 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

%t Table[Mod[n!+2,n+2],{n,0,100}] (* _Harvey P. Dale_, Sep 01 2022 *)

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

%o (PARI) A249355(n)=if(n>2,isprime(n+2)+2,!n) \\ _M. F. Hasler_, Oct 31 2014

%o (Magma) [(Factorial(n)+2) mod(n+2): n in [0..100]]; // _Vincenzo Librandi_, Oct 27 2014

%Y Cf. A055976, A242707.

%K nonn

%O 0,4

%A _M. F. Hasler_, Oct 26 2014