cp's OEIS Frontend

A260232 Least prime p such that pi(p*n) = n*pi(q*n) for some prime q.

%I A260232 #21 Aug 02 2015 04:07:30
%S A260232 2,5,13,67,23,19,433,443,107,41,61,251,239,1987,541,491,1093,499,421,
%T A260232 179,2137,1297,1097,101,103,2411,1283,1847,379,4993,8329,5563,4297,
%U A260232 5639,9587,1867,5113,6691,3691,1193,4663,2971,27733,7121,593,2273,607,6047,4217,2609
%N A260232 Least prime p such that pi(p*n) = n*pi(q*n) for some prime q.
%C A260232 Conjecture: For any positive integer n, each rational number r > 0 can be written as pi(p*n)/pi(q*n) with p and q both prime.
%C A260232 For example, 4/7 = 416/728 = pi(479*6)/pi(919*6) with 479 and 919 both prime.
%C A260232 The conjecture holds trivially for n = 1 since pi(prime(m)*1) = m for all m = 1,2,3,.... Also, the conjecture implies that a(n) exists for any n > 0.
%D A260232 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 A260232 Zhi-Wei Sun, <a href="/A260232/b260232.txt">Table of n, a(n) for n = 1..300</a>
%H A260232 Zhi-Wei Sun, <a href="/A260232/a260232_2.txt">Checking the conjecture for n = 1..10 and r = a/b with a,b = 1..100</a>
%H A260232 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 A260232 a(4) = 67 since pi(67*4) = 56 = 4*14 = 4*pi(11*4) with 11 and 67 both prime.
%t A260232 f[n_]:=PrimePi[n]; Do[k=0;Label[bb];k=k+1;If[Mod[f[Prime[k]*n],n]>0,Goto[bb]];Do[If[f[Prime[k]n]==n*f[Prime[j]*n],Goto[aa]];If[f[Prime[k]n]<n*f[Prime[j]*n],Goto[bb]];Continue,{j,1,k}];Label[aa];Print[n," ",Prime[k]];Continue,{n,1,50}]
%Y A260232 Cf. A000040, A000720, A257364, A257928, A260140, A260252.
%K A260232 nonn
%O A260232 1,1
%A A260232 _Zhi-Wei Sun_, Jul 20 2015