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A181901 a(n) = least positive integer m such that 2(s_k)^2 for k=1,...,n are pairwise distinct modulo m where s_k = Sum_{j=1..k} (-1)^(k-j)*p_j and p_j is the j-th prime.

%I A181901 #25 Jun 22 2020 19:20:41
%S A181901 1,4,7,9,13,17,19,23,25,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
%T A181901 101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,
%U A181901 191,193,197,199,211,223,227,229,233
%N A181901 a(n) = least positive integer m such that 2(s_k)^2 for k=1,...,n are pairwise distinct modulo m where s_k = Sum_{j=1..k} (-1)^(k-j)*p_j and p_j is the j-th prime.
%C A181901 On Mar 28 2012, _Zhi-Wei Sun_ conjectured that a(n) is the (n+1)-th prime p_{n+1} with the only exceptions being a(1)=1, a(2)=4, a(4)=9 and a(9)=25. He has shown that 2(s_k)^2 (k=1,...,n) are indeed pairwise distinct modulo p_{n+1} and hence a(n) does not exceed p_{n+1}.
%C A181901 Note that the sequence 0,s_1,s_2,s_3,... is A008347 introduced by _N. J. A. Sloane_ and _J. H. Conway_.
%C A181901 Compare a(n) with the sequence A210640.
%C A181901 The conjecture was verified for n up to 2*10^5 by the author in 2018, and for n up to 3*10^5 by Chang Zhang (a student at Nanjing Univ.) in June 2020. - _Zhi-Wei Sun_, Jun 22 2020
%H A181901 Zhi-Wei Sun, <a href="/A181901/b181901.txt">Table of n, a(n) for n = 1..600</a>
%H A181901 Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;95333354.1203">An amazing recurrence for primes</a>, a message to Number Theory List, March 31, 2012.
%H A181901 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 A181901 We have a(4)=9 since 2(s_1)^2=8, 2(s_2)^2=2, 2(s_3)^2=32, 2(s_4)^2=18 are pairwise distinct modulo 9 but not pairwise distinct modulo any of 1,...,8.
%t A181901 s[n_]:=Sum[(-1)^k*Prime[k],{k,1,n}]
%t A181901 f[n_]:=2*s[n]^2
%t A181901 R[n_,m_]:=Union[Table[Mod[f[k],m],{k,1,n}]]
%t A181901 Do[Do[If[Length[R[n,m]]==n,Print[n," ",m];Goto[aa]],{m,1,Prime[n+1]}];
%t A181901    Print[n];Label[aa];Continue,{n,1,600}]
%Y A181901 Cf. A000040, A008347, A210640, A210393, A210394, A210186, A210144, A208494, A208643, A207982.
%K A181901 nonn
%O A181901 1,2
%A A181901 _Zhi-Wei Sun_, Mar 31 2012