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%I #36 Nov 16 2020 23:39:44
%S 2,3,5,7,11,19,23,23,23,47,59,61,71,71,71,101,101,101,101,101,101,113,
%T 113,113,113,113,113,113,113,113,223,223,223,223,223,223,223,223,223,
%U 223,223,223,223,223,223,223,223,223,487,487,661,661,661,661,661,661,661,661,661,719,719,719,719,719,719,811,811,811,811,811,811,811,811,811,811
%N a(n) = least integer m>1 such that m divides none of P_i + P_j with 0<i<j<=n where P_k is the product of the first k primes.
%C Conjecture: all the terms are primes and a(n) < n^2 for all n > 1.
%H Zhi-Wei Sun, <a href="/A210186/b210186.txt">Table of n, a(n) for n = 1..258</a>
%H Romeo Meštrović, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018. - From _N. J. A. Sloane_, Jun 13 2012
%H Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;df748f41.1202">A function taking only prime values</a>, message to Number Theory List, Feb. 21, 2012.
%H 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, Vol. 133, No. 8 (2013), pp. 2794-2812.
%e We have a(3)=5 since 2+2*3, 2+2*3*5, 2*3+2*3*5 are pairwise distinct modulo m=5 but not pairwise distinct modulo m=2,3,4.
%t P[n_]:=Product[Prime[k],{k,1,n}]
%t R[n_,m_]:=Product[If[Mod[P[k]+P[j],m]==0,0,1],{k,2,n},{j,1,k-1}]
%t Do[Do[If[R[n,m]==1,Print[n," ",m];Goto[aa]],{m,2,Max[2,n^2]}]; Print[n];Label[aa];Continue,{n,1,300}]
%Y Cf. A000040, A210144, A208494, A208643, A207982.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Mar 18 2012
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