A255677 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.

5, 30, 8458, 18, 252, 25, 1407, 476, 9098, 108, 1814, 1868, 153, 1005, 67, 26532, 1592, 200, 963, 99, 833, 1356, 3869, 981, 531, 127, 4961, 366, 1192, 1873, 41308, 409, 21756, 194664, 180, 27071, 7433, 160179, 2076, 544, 211, 10639, 19571, 33483, 603, 68380, 1517, 47529, 35923

Offset: 2

Author

Zhi-Wei Sun, Jul 10 2015

Keywords

• nonn

Comments

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.

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.

Examples

			a(2) = 5 since pi(5)^2 + pi(5*2)^2 = 3^2 + 4^2 = 5^2.
a(3) = 30 since pi(30)^2 + pi(30*3)^2 = 10^2 + 24^2 = 26^2.
a(68) = 6260592 since pi(6260592)^2 + pi(6260592*68)^2 = 429505^2 + 22632876^2 = 22636951^2.
a(95) = 7955506 since pi(7955506)^2 + pi(7955506*95)^2 = 536984^2 + 38985687^2 = 38989385^2.
		

References

• 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.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 2..100
• Zhi-Wei Sun, Checking the conjecture for r = a/b, b/a with 1 <= a < b <= 50
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

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

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
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}]

PARI

a(n)={ k=2; while(!issquare(primepi(k)^2 + primepi(k*n)^2),k++); return(k);}
main(size)={ v=vector(size); for(i=2, size+1, v[i-1]=a(i)); return(v);} /* Anders Hellström, Jul 11 2015 */