Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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