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Least positive integer k < n with k!*(n-k) + 1 prime, or 0 if such an integer k does not exist.
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%I #12 Apr 19 2019 16:58:14

%S 0,1,1,2,1,3,1,2,3,2,1,4,1,3,3,2,1,4,1,2,3,2,1,3,2,3,6,2,1,3,1,2,3,6,

%T 2,3,1,2,6,3,1,5,1,6,5,2,1,3,3,2,4,2,1,3,2,2,6,2,1,11,1,5,5,3,2,3,1,5,

%U 3,2,1,6,1,7,3,2,2,4,1,2,6,4,1,3,2,3,4,2,1,3,2,2,3,3,6,7,1,2,3,2

%N Least positive integer k < n with k!*(n-k) + 1 prime, or 0 if such an integer k does not exist.

%C Conjecture: 0 < a(n) < sqrt(n)*(log n) for all n > 2.

%C See also the conjecture in A231516.

%H Zhi-Wei Sun, <a href="/A231631/b231631.txt">Table of n, a(n) for n = 1..10000</a>

%e a(4) = 2 since 1!*3 + 1 = 4 is not prime, but 2!*2 + 1 = 5 is prime.

%t Do[Do[If[PrimeQ[x!*(n-x)+1],Print[n," ",x];Goto[aa]],{x,1,n-1}];

%t Print[n," ",0];Label[aa];Continue,{n,1,100}]

%t lpik[n_]:=Module[{k=1},While[!PrimeQ[k!(n-k)+1],k++];k]; Join[{0},Array[ lpik,100,2]] (* _Harvey P. Dale_, Apr 19 2019 *)

%Y Cf. A000040, A000142, A231201, A231516, A231555, A231561, A231557

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Nov 11 2013