OFFSET
1,2
COMMENTS
For all a(n) > 1, there is only one k that makes (k!+n)/(k+n) a prime (up to k <= 5000). For a(2) through a(11), they are given by {4, 6, 4, 12, 5, 6, 6, 12, 6, 44} respectively.
From Robert Israel, Aug 09 2016: (Start)
If n > 1 and (k!+n)/(k+n) is prime, we must have k < n.
Proof: if k >= n, then n | k! + n, so (after eliminating some small cases) for (k!+n)/(k+n) to be prime we need n | k+n and thus n | k. But if k = mn,
mn-1 >= m+1 so m+1 | (mn-1)!, and (k!+n)/(k+n)=(m(mn-1)!+1)/(m+1) is not an integer. (End)
EXAMPLE
(k!+6)/(k+6) is prime for some k (let k = 4). Thus 6 is a member of this sequence.
MAPLE
f:= proc(n) ormap(proc(k) local v; v:= (k!+n)/(k+n); v::integer and isprime(v) end proc, [$2..n-1]) end proc:
f(1):= true:
select(f, [$1..1500]); # Robert Israel, Aug 09 2016
MATHEMATICA
fQ[n_] := Block[{k = 1}, While[k < n && !PrimeQ[(k! + n)/(k + n)], k++]; k < n]; k = 2; lst = {1}; While[k < 25001, If[fQ@k, Print@k; AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Aug 14 2016 *)
PROG
(PARI) a(n)=for(k=1, 5000, s=(k!+n)/(k+n); if(floor(s)==s, if(ispseudoprime(s), return(k))))
n=1; while(n<1000, if(a(n), print1(n, ", ")); n+=1)
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Derek Orr, May 26 2014
EXTENSIONS
a(12)-a(15) from Robert Israel, Aug 09 2016
a(16)-a(21) from Robert G. Wilson v, Aug 11 2016
a(22) from Robert G. Wilson v, Aug 13 2016
STATUS
approved