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A222114 Least integer m>1 such that 6*p_k*(p_k-1) (k=1,...,n) are pairwise incongruent modulo m, where p_k denotes the k-th prime.

%I A222114 #13 Dec 06 2018 16:34:59
%S A222114 2,5,5,13,19,29,31,37,37,37,61,61,61,89,97,97,97,109,131,139,149,157,
%T A222114 157,157,173,181,193,193,193,193,241,241,241,271,271,271,271,317,331,
%U A222114 331,331,349,349,367,367,367,397,397,397,397,397,397,457,457,457,457,457,457,523,523
%N A222114 Least integer m>1 such that 6*p_k*(p_k-1) (k=1,...,n) are pairwise incongruent modulo m, where p_k denotes the k-th prime.
%C A222114 Conjecture: For each n=3,4,...,  a(n) is the first prime p>=p_n dividing none of those p_i+p_j-1 (1<=i<j<=n).
%H A222114 Zhi-Wei Sun, <a href="/A222114/b222114.txt">Table of n, a(n) for n = 1..10000</a>
%H A222114 Zhi-Wei Sun, <a href="https://doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory, 133 (2013), 2794-2812.
%e A222114 a(2)=5 since 6*p_1*(p_1-1)=12 and 6*p_2*(p_2-1)=36 are incongruent modulo 5 but 12 is congruent to 36 modulo any of 2, 3, 4.
%t A222114 R[n_,m_]:=Union[Table[Mod[6Prime[k](Prime[k]-1),m],{k,1,n}]]
%t A222114 s=2
%t A222114 Do[Do[If[Length[R[n,m]]==n,s=m;Print[n," ",m];Goto[aa]],{m,s,n^2}];
%t A222114 Print[n];Label[aa];Continue,{n,1,100}]
%Y A222114 Cf. A000040, A208643, A181901.
%K A222114 nonn
%O A222114 1,1
%A A222114 _Zhi-Wei Sun_, May 13 2013