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A257364 Least prime p such that pi(p*n)^2 = pi(q*n)^2 + pi(r*n)^2 for some primes q and r, where pi(x) denotes the number of primes not exceeding x.

%I A257364 #30 Jul 28 2015 07:54:12
%S A257364 11,59,47,211,23,233,181,257,109,109,13,311,929,47,389,757,1747,13,67,
%T A257364 2389,1087,569,311,853,103,5569,1399,3203,10891,3673,3793,1873,4357,
%U A257364 41,2297,131,3253,6737,2621,5113,2879,953,6379,3539,12343,4337,6067,11939,43441,5179
%N A257364 Least prime p such that pi(p*n)^2 = pi(q*n)^2 + pi(r*n)^2 for some primes q and r, where pi(x) denotes the number of primes not exceeding x.
%C A257364 Conjecture: a(n) exists for any n > 0. In other words, for each fixed positive integer n the sequence pi(p*n) with p prime contains a Pythagorean triple.
%C A257364 This is stronger than the conjecture in A255679.
%D A257364 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 A257364 Zhi-Wei Sun, <a href="/A257364/b257364.txt">Table of n, a(n) for n = 1..100</a>
%H A257364 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 A257364 a(1) = 11 since 5, 7 and 11 are primes with pi(5*1)^2 + pi(7*1)^2 = 3^2 + 4^2 = 5^2 = pi(11*1)^2.
%e A257364 a(45) = 12343 since 4337, 11311 and 12343 are primes with pi(4337*45)^2 + pi(11311*45)^2 = 17590^2 + 42216^2 = 45734^2 = pi(12343*45)^2.
%e A257364 a(49) = 43441 since 15427, 39839 and 43441 are primes with pi(15427*49)^2 + pi(39839*49)^2 = 60685^2 + 145644^2 = 157781^2 = pi(43441*49)^2.
%t A257364 f[n_]:=PrimePi[n]
%t A257364 Do[k=0;Label[bb];k=k+1;Do[Do[If[f[Prime[k]*n]^2==f[Prime[i]*n]^2+f[Prime[j]*n]^2,Goto[aa]];If[f[Prime[k]*n]^2<f[Prime[i]*n]^2+f[Prime[j]*n]^2,Goto[cc]];Continue,{i,1,j-1}];Label[cc];Continue,{j,1,k-1}];Goto[bb];
%t A257364 Label[aa];Print[n," ",Prime[k]];Continue,{n,1,100}]
%Y A257364 Cf. A000040, A000720, A255679, A257928.
%K A257364 nonn
%O A257364 1,1
%A A257364 _Zhi-Wei Sun_, Jul 11 2015