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%I #10 Jan 04 2018 11:55:29
%S 1,3,4,11,7,7,10,17,43,13,51,22,51,36,31,49,64,71,119,73,86,68,141,
%T 110,153,85,83,86,144,81,174,127,115,87,122,138,143,134,133,142,211,
%U 229,152,104,109,177,259,142,194,176,196,311,312,193,243,197,396,169,156,171
%N Least positive integer m <= n^2/2 + 3 such that {k*prime(k): k = 1,...,m} contains a complete system of residues modulo n, or 0 if such a number m does not exist.
%C Conjecture: (i) a(n) > 0 for all n > 0.
%C (ii) For any positive integer n not equal to 3, the number n*prime(n) + 1 cannot be a power x^m with m > 1.
%C (iii) There are infinitely many positive integers n with n - 1, n + 1, n + prime(n), n + prime(n)^2, n^2 + prime(n), n^2 + prime(n)^2 all prime.
%H Chai Wah Wu, <a href="/A232891/b232891.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..1000 from Zhi-Wei Sun)
%e a(3) = 4 since 1*prime(1) = 2, 2*prime(2) === 3*prime(3) == 0 (mod 3), and 4*prime(4) = 28 == 1 (mod 3).
%t L[m_,n_]:=Length[Union[Table[Mod[k*Prime[k],n],{k,1,m}]]]
%t Do[Do[If[L[m,n]==n,Print[n," ",m];Goto[aa]],{m,1,n^2/2+3}];
%t Print[n," ",counterexample];Label[aa];Continue,{n,1,60}]
%Y Cf. A000040, A232616, A232861, A232862.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Dec 02 2013
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