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%I #9 Aug 25 2015 05:45:59
%S 1,755,4648,1335,1096,14708,5964,636,1063,13019,9808,2776,2580,2797,
%T 6421,1573,2432,4790,862,1855,566,2145,18554,35634,5264,1293,39402,
%U 1445,2397,17930,586,2526,24571,18403,5480,366,5159,9710,179,4469,6757,7866,263,1701,2941,477,5032,10705,3494,8597,953,11954,2586,689,9456,1309,8651,12538,4106,13762
%N Least positive integer k such that prime(prime(prime(k)))+ prime(prime(prime(k*n))) = 2*prime(prime(p)) for some prime p.
%C Conjecture: (i) Any positive rational number r can be written as m/n, where m and n are positive integers such that (prime(prime(prime(m))) + prime(prime(prime(n))))/2 = prime(prime(p)) for some prime p.
%C (ii) Let p(1,n) = prime(n), and p(m+1,n) = p(m,prime(n)) for m,n = 1,2,3,.... Then, for any integers m > 0 and k > 2, the sequence p(m,n) (n = 1,2,3,...) contains infinitely many nontrivial k-term arithmetic progressions.
%C (iii) Let m be any positive integer. Then the sequence p(m,n)^(1/n) (n = 1,2,3,...) is strictly decreasing. Also, for any relatively prime integers q > 0 and r, there are infinitely many n > 0 such that p(m,n) == r (mod q).
%C Note that part (ii) of the conjecture extends the Green-Tao theorem and the third part of the conjecture in A261462. Also, part (iii) in the case m = 1 reduces to the Firoozbakht conjecture on primes and Dirichlet's theorem on primes in arithmetic progressions.
%D Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H Zhi-Wei Sun, <a href="/A261583/b261583.txt">Table of n, a(n) for n = 1..200</a>
%H Zhi-Wei Sun, <a href="/A261583/a261583.txt">Checking part (i) of the conjecture for r = a/b with a,b = 1..100</a>
%H B. Green and T. Tao, <a href="http://dx.doi.org/10.4007/annals.2008.167.481 ">The primes contain arbitrary long arithmetic progressions</a>, Annals of Math. 167(2008), 481-547.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e a(2) = 755 since prime(prime(prime(755))) + prime(prime(prime(755*2))) = prime(prime(5741)) + prime(prime(12641)) = prime(56611) + prime(135671) = 700897 + 1808581 = 2*1254739 = 2*prime(96797) = 2*prime(prime(9319)) with 9319 prime.
%t f[n_]:=Prime[Prime[Prime[n]]]
%t PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]&&PrimeQ[PrimePi[PrimePi[p]]]
%t Do[k=0;Label[bb];k=k+1;If[PQ[(f[k]+f[k*n])/2],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]
%Y Cf. A000040, A006450, A261437, A261462.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Aug 25 2015
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