A255677 - OEIS

%I #33 Jul 16 2015 08:37:20

%S 5,30,8458,18,252,25,1407,476,9098,108,1814,1868,153,1005,67,26532,

%T 1592,200,963,99,833,1356,3869,981,531,127,4961,366,1192,1873,41308,

%U 409,21756,194664,180,27071,7433,160179,2076,544,211,10639,19571,33483,603,68380,1517,47529,35923

%N Least integer k > 1 such that pi(k)^2 + pi(k*n)^2 is a square, where pi(.) is the prime-counting function given by A000720.

%C Conjecture: Each positive rational number r < 1 can be written as m/n with 1 < m < n such that pi(m)^2 + pi(n)^2 is a square. Also, any rational number r > 1 can be written as m/n with m > n > 1 such that pi(m)^2 - pi(n)^2 is a square.

%C For example, 23/24 = 19947716/20815008 with pi(19947716)^2 + pi(20815008)^2 = 1267497^2 + 1319004^2 = 1829295^2, and 7/3 = 26964/11556 with pi(26964)^2 - pi(11556)^2 = 2958^2 - 1392^2 = 2610^2.

%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="/A255677/b255677.txt">Table of n, a(n) for n = 2..100</a>

%H Zhi-Wei Sun, <a href="/A255677/a255677_1.txt">Checking the conjecture for r = a/b, b/a with 1 <= a < b <= 50</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(2) = 5 since pi(5)^2 + pi(5*2)^2 = 3^2 + 4^2 = 5^2.

%e a(3) = 30 since pi(30)^2 + pi(30*3)^2 = 10^2 + 24^2 = 26^2.

%e a(68) = 6260592 since pi(6260592)^2 + pi(6260592*68)^2 = 429505^2 + 22632876^2 = 22636951^2.

%e a(95) = 7955506 since pi(7955506)^2 + pi(7955506*95)^2 = 536984^2 + 38985687^2 = 38989385^2.

%t SQ[n_]:=IntegerQ[Sqrt[n]]

%t Do[k=1;Label[aa];k=k+1;If[SQ[PrimePi[k]^2+PrimePi[k*n]^2],Goto[bb],Goto[aa]];Label[bb];Print[n," ",k];Continue,{n,2,50}]

%o (PARI) a(n)={ k=2; while(!issquare(primepi(k)^2 + primepi(k*n)^2),k++); return(k);}

%o main(size)={ v=vector(size); for(i=2, size+1, v[i-1]=a(i)); return(v);} /* _Anders Hellström_, Jul 11 2015 */

%Y Cf. A000720, A000290, A255679, A259531, A259789.

%K nonn

%O 2,1

%A _Zhi-Wei Sun_, Jul 10 2015