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%I #17 Jul 14 2015 02:25:29
%S 13,7,13,67,19,79,47,193,107,41,229,179,383,281,173,1327,193,701,1429,
%T 211,113,73,1093,83,1447,659,197,719,331,761,1171,2269,467,509,863,
%U 113,643,577,563,379,607,1291,283,3593,2549,881,1523,4663,2657,3583,8807,683,2251,863,8929,163,6737,2459,4919,6553
%N Least prime p such that pi(p*n) = pi(q*n)*pi(r*n) for some primes q and r with p > q > r, where pi(x) denotes the number of primes not exceeding x.
%C Conjecture: a(n) exists for any n > 0. Also, for each positive integer n there are distinct primes p, q and r such that pi(p*n) = pi(q*n) + pi(r*n).
%D 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 Zhi-Wei Sun, <a href="/A257928/b257928.txt">Table of n, a(n) for n = 1..200</a>
%H 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 a(1) = 13 since 3, 5 and 13 are distinct primes with pi(13*1) = 6 = 2*3 = pi(3*1)*pi(5*1).
%e a(200) = 105227 since 19, 113 and 105227 are distinct primes with pi(105227*200) = 1332672 = 528*2524 = pi(19*200)*pi(113*200).
%t f[n_]:=PrimePi[n]
%t Do[k=0;Label[bb];k=k+1;Do[Do[If[f[Prime[k]*n]==f[Prime[i]*n]*f[Prime[j]*n],Goto[aa]];If[f[Prime[k]*n]<f[Prime[i]*n]*f[Prime[j]*n],Goto[cc]];Continue,{i,1,j-1}];Label[cc];Continue,{j,1,k-1}];Goto[bb];
%t Label[aa];Print[n," ",Prime[k]];Continue,{n,1,60}]
%o (PARI) a(n)={my(i,j,k=3);while(1,for(j=2,k-1,for(i=1,j-1,if(primepi(prime(k)*n) == primepi(prime(i)*n)*primepi(prime(j)*n),break(3));));k++);return(prime(k));} main(size)={return(vector(size,n,a(n)));} /* _Anders Hellström_, Jul 13 2015 */
%Y Cf. A000040, A000720, A257364, A257938.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jul 13 2015
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