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A260888 Least prime p such that 2 + 3*pi(p*n) = 4*pi(q*n) for some prime q, where pi(x) denotes the number of primes not exceeding x.

%I A260888 #9 Aug 03 2015 04:21:22
%S A260888 3,2,41,211,23,83,43,23,7,3,601,109,23,251,31,251,7,41,149,157,293,3,
%T A260888 103,41,2083,233,7,647,1877,7,1117,599,7,937,487,7,251,149,7,439,83,3,
%U A260888 7,43,643,7,157,157,1291,7
%N A260888 Least prime p such that 2 + 3*pi(p*n) = 4*pi(q*n) for some prime q, where pi(x) denotes the number of primes not exceeding x.
%C A260888 Conjecture: Let a and b be relatively prime positive integers, and let c be any integer. For any positive integer n, there are primes p and q such that a*pi(p*n) - b*pi(q*n) = c.
%C A260888 In the case c = 0, this reduces to the conjecture in A260232.
%C A260888 For example, for a = 20, b = 19, c = 18 and n = 28, we have 20*pi(4549*28)-19*pi(4813*28) = 20*11931-19*12558 = 18 with 4549 and 4813 both prime.
%H A260888 Zhi-Wei Sun, <a href="/A260888/b260888.txt">Table of n, a(n) for n = 1..600</a>
%H A260888 Zhi-Wei Sun, <a href="/A260888/a260888.txt">Checking the conjecture for a,b,c = 1..20 and n = 1..30</a>
%H A260888 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1585">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2015.
%e A260888 a(5) = 23 since 2+3*pi(23*5) = 2+3*30 = 92 = 4*23 = 4*pi(17*5) with 23 and 17 both prime.
%t A260888 f[k_,n_]:=PrimePi[Prime[k]*n]
%t A260888 Do[k=0;Label[bb];k=k+1;If[Mod[3*f[k,n]+2,4]>0,Goto[bb]];Do[If[(3*f[k,n]+2)/4==f[j,n],Goto[aa]];If[(3*f[k,n]+2)/4<f[j,n],Goto[bb]];Continue,{j,1,k}];Label[aa];Print[n," ", Prime[k]];Continue,{n,1,50}]
%Y A260888 Cf. A000040, A000720, A260140, A260232, A260886.
%K A260888 nonn
%O A260888 1,1
%A A260888 _Zhi-Wei Sun_, Aug 02 2015