OFFSET
1,1
COMMENTS
The entries a(n) = 0 are based on checking values of k <= 20000. - Derek Orr, May 21 2014
a(33) > 456000. - Derek Orr, Oct 23 2014 and Jon E. Schoenfield, Oct 25 2014
If a(n) > n, a(n) + n is necessarily prime.
For n = 33..100, the sequence continues with a(33), 12, 36, 6, 16, 36, 12, 12, 42, 40, 54, 42, 6, 16, 340730, 46, 774, 99, 90196, 16, 44, 52, 18, 12, 12, 1249, a(59), 9, a(61), 60, 30, 169, 43346, 12, 22, 38, 70, a(70), 2352, 70, a(73), 72, 36, 123183, a(77), 1283, a(79), 12, 118, a(82), 84, 82, a(85), 23, 42, 28, 110, 12, a(91), a(92), 883964, a(94), 14888, 6, 726, 96, 232, 10. For each n in {33, 59, 61, 70, 73, 77, 79, 82, 85, 91, 92, 94}, a(n) > 2*10^6.
a(33) > 4*10^6. - Jon E. Schoenfield, Apr 30 2018
EXAMPLE
(3!+1)/(3+1) = 7/4 is not an integer. (4!+1)/(4+1) = 25/5 = 5 is an integer. So a(1) = 4.
a(2) = 0, because k+2 can never divide k!+2: If k+2 = p > 3 is a prime, then k!+2 = (p-2)!+2 == (1+2) (mod p), using the fact that (p-2)! == 1 (mod p) for all primes p. If k+2 > 4 is composite, let k+2 = p*m for some prime p and some m > 2. Then p*m-2 >= m and also p*m-2 > p. Thus, k! = (p*m-2)! is divisible by p*m. (If m = p this still holds because then p*m-2 >= 2*m.) Therefore, k!+2 == 2 (mod k+2) in this case.
a(3) = 0 for similar reasons: Consider s = (k!+3) mod (k+3). If k+3 is prime, s = (k+8)/2 and if k+3 is composite, s = 3. So s is never 0.
a(4) = 0 as well: Let s = (k!+4) mod (k+4). If k+4 is prime, there are two cases. If k == 1 (mod 6), then s = (k+29)/6. If k == 3 (mod 6), then s = (5*k+45)/6 except at the following finite number of points (k,s): (3,3), (9,2), and (15,1). If k+4 is composite, s = 4 except at (k,s) = (5,7). Thus, s is never 0. - Jon E. Schoenfield, Oct 28 2014
PROG
(PARI) a(n)=k=3; while((k!+n)%(k+n)&&k<2e6, k++); k \\ program improved by Derek Orr, Oct 23 2014
(PARI) a(n)=for(k=3, n, if((k!+n)%(k+n)==0, return(k))); forprime(kn=2*n, 2e6+n, if(prod(i=2, kn-n, i, Mod(1, kn))==-n, return(kn-n))); 0 \\ Charles R Greathouse IV, Oct 28 2014
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Derek Orr, May 21 2014
EXTENSIONS
More terms added by Derek Orr, Oct 23 2014 and Jon E. Schoenfield, Oct 25 2014
Edited by M. F. Hasler, Oct 30 2014
STATUS
approved