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%I #14 Aug 03 2015 09:53:08
%S 2,157,199,3539,1973,9241,14629,167,48281,2207,313,30631,35993,33863,
%T 23,23,7963,17077,11069,6043,4931,3697,2339,14153,35311,63149,111143,
%U 491,247193,464237,2293,12101,727,61403,243437,40289,4337,241,2719,13933,21817,6803,52813,451279,166409,45631,109891,490969,153563,9127
%N Least prime p such that 3 + 4*prime(p*n) = 5*prime(q*n) for some prime q.
%C Conjecture: Let a,b,c be pairwise relatively prime positive integers with a+b+c even and a not equal to b. Then, for any positive integer n, there are primes p and q such that a*prime(p*n) - b*prime(q*n) = c.
%C This includes the conjectures in A260252 and A260882 as special cases.
%C For example, for a = 7, b = 17, c = 20 and n = 30, we have 7*prime(4695851*30) - 17*prime(2020243*30) = 7*2922043519 - 17*1203194389 = 20 with 4695851 and 2020243 both prime.
%H Zhi-Wei Sun, <a href="/A260886/b260886.txt">Table of n, a(n) for n = 1..200</a>
%H Zhi-Wei Sun, <a href="/A260886/a260886.txt">Checking the conjecture for a,b,c = 1..20 and n = 1..30</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2015.
%e a(2) = 157 since 3 + 4*prime(157*2) = 3 + 4*2083 = 8335 = 5*prime(131*2) with 157 and 131 both prime.
%t f[n_]:=Prime[n]
%t PQ[p_,n_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]
%t Do[k=0;Label[bb];k=k+1;If[PQ[(4*f[n*f[k]]+3)/5,n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", f[k]];Continue,{n,1,50}]
%Y Cf. A000040, A260120, A260252, A260882.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Aug 02 2015
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