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%I #15 Aug 28 2014 09:20:55
%S 2,2,17,4,1,1,59,3,56,1,39,10,9,130,2,18,11,7,80,67,2,19,27,17,92,73,
%T 180,65,5,110,282,4,6,8,16,2,23,198,20,3,99,83,217,13,110,28,16,6,5,3,
%U 144,31,9,187,176,145,75,11,43,424,4,54,272,8,26,131,123,107,8,4,52,9,127,84,264,33,145,663,16,285
%N Least positive integer k such that prime(k*n), prime((k+1)*n) and prime((k+2)*n) form an arithmetic progression, or 0 if such a number k does not exist.
%C Conjecture: (i) We always have 0 < a(n) < 3*prime(n) + 9.
%C (ii) For any integer n > 3, there is a positive integer k < 5*n^3 such that prime(k*n), prime((k+1)*n), prime((k+2)*n), and prime((k+3)*n) form a 4-term arithmetic progression.
%C (iii) In general, for each m = 3, 4, ..., if n is sufficiently large then there is a positive integer k = O(n^(m-1)) such that prime((k+j)*n) (j = 0, ..., m-1) form an arithmetic progression.
%C The conjecture is a refinement of the Green-Tao theorem.
%H Zhi-Wei Sun, <a href="/A238289/b238289.txt">Table of n, a(n) for n = 1..8000</a>
%H B. Green and T. Tao, <a href="http://dx.doi.org/10.4007/annals.2008.167.481">The primes contain arbitrarily long arithmetic progressions</a>, Annals of Math. 167(2008), 481-547.
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e a(2) = 2 since prime(2*2) = 7, prime(3*2) = 13 and prime(4*2) = 19 form a 3-term arithmetic progression, but prime(1*2) = 3, prime(2*2) = 7 and prime(3*2) = 13 do not form a 3-term arithmetic progression.
%t d[k_,n_]:=Prime[(k+1)*n]-Prime[k*n]
%t Do[Do[If[d[k,n]==d[k+1,n],Print[n," ",k];Goto[aa]],{k,1,3*Prime[n]+8}];
%t Print[n," ",0];Label[aa];Continue,{n,1,80}]
%o (PARI) okpr(p, q, r) = (q - p) == (r - q);
%o a(n) = {k = 1; while(! okpr(prime(k*n), prime((k+1)*n), prime((k+2)*n)), k++); k;} \\ _Michel Marcus_, Aug 28 2014
%Y Cf. A000040, A238277, A238278, A238281.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Feb 22 2014
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