cp's OEIS Frontend

A261462 Least positive integer k such that prime(prime(k)), prime(prime(k*n)), prime(p) and prime(q) form a 4-term arithmetic progression for some pair of primes p and q.

%I A261462 #14 Aug 29 2015 17:25:28
%S A261462 1,5109,879,27956,103840,32205,21404,1800,4241,81794,2355,4352,14974,
%T A261462 8552,26159,17621,91986,52574,73764,4699,12546,27347,71148,6819,333,
%U A261462 38830,28809,2058,24609,84,11478,226251,21383,54,2930,36423,2602,22,47668,15594,19,56106,41913,72620,211070,9022,2587,10316,74965,26852
%N A261462 Least positive integer k such that prime(prime(k)), prime(prime(k*n)), prime(p) and prime(q) form a 4-term arithmetic progression for some pair of primes p and q.
%C A261462 Conjecture: (i) Any positive rational number r can be written as m/n, where m and n are positive integers such that (prime(prime(m))+prime(prime(n)))/2 = prime(p) for some prime p.
%C A261462 (ii) Any positive rational number r <= 1 can be written as m/n, where m and n are positive integers such that prime(prime(m)),prime(prime(n)), prime(p) and prime(q) form a 4-term arithmetic progression for some pair of primes p and q.
%C A261462 (iii) For any integer k > 2, the set {prime(p): p is prime} contains infinitely many nontrivial k-term arithmetic progressions.
%D A261462 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 A261462 Zhi-Wei Sun, <a href="/A261462/b261462.txt">Table of n, a(n) for n = 1..100</a>
%H A261462 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 A261462 Zhi-Wei Sun, <a href="/A261462/a261462.txt">Checking part (ii) of the conjecture for r = a/b with 1 <= a <= b <= 50</a>
%H A261462 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 A261462 a(2) = 5109 since prime(prime(5109)) = 608591, prime(prime(5109*2)) = 1401791, prime(162343) = 2194991, and prime(216023) = 2988191 form a 4-term arithmetic progression with 162343 and 216023 both prime.
%t A261462 f[n_]:=Prime[Prime[n]]
%t A261462 PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
%t A261462 Do[k=0;Label[bb];k=k+1;If[PQ[2*f[k*n]-f[k]]&&PQ[3*f[k*n]-2*f[k]],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,50}]
%Y A261462 Cf. A000040, A006450, A232502, A238289, A261385, A261395.
%K A261462 nonn
%O A261462 1,2
%A A261462 _Zhi-Wei Sun_, Aug 20 2015