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A210393 a(n) = least integer m>1 such that S_k! for k=1,...,n are pairwise distinct modulo m where S_k is the sum of the first k primes.

%I A210393 #22 Apr 18 2013 02:58:27
%S A210393 2,3,7,13,19,29,43,61,79,101,131,167,199,239,293,331,389,443,503,571,
%T A210393 641,719,797,877,971,1063,1163,1277,1373,1481,1601,1721,1861,1997,
%U A210393 2131,2281,2437,2591,2753,2927,3089,3271,3457,3659,3847,4049,4231,4441,4663,4889
%N A210393 a(n) = least integer m>1 such that S_k! for k=1,...,n are pairwise distinct modulo m where S_k is the sum of the first k primes.
%C A210393 When n>1, we have S_n!=S_{n-1}!=0 (mod m) for all m=1,...,S_{n-1} and hence a(n)>S_{n-1}. Zhi-Wei Sun conjectured that a(n) is always a prime not exceeding S_n.
%H A210393 Zhi-Wei Sun, <a href="/A210393/b210393.txt">Table of n, a(n) for n = 1..720</a>
%H A210393 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory 133(2013), no.8, 2794-2812.
%e A210393 a(3)=7 since 2!,(2+3)!,(2+3+5)! are pairwise distinct modulo m=7 but not pairwise distinct modulo m=2,3,4,5,6.
%t A210393 s[n_]:=s[n]=Sum[Prime[k],{k,1,n}]
%t A210393 f[n_]:=f[n]=s[n]!
%t A210393 R[n_,m_]:=Union[Table[Mod[f[k],m],{k,1,n}]]
%t A210393 Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,Max[2,s[n-1]],s[n]}];
%t A210393    Print[n];Label[aa];Continue,{n,1,720}]
%Y A210393 Cf. A000040, A210394, A210186, A210144, A208494, A208643, A207982.
%K A210393 nonn
%O A210393 1,1
%A A210393 _Zhi-Wei Sun_, Mar 20 2012