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

2,1

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.

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.

Example

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.

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}]

Prog

(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 */

Crossrefs

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

Sequence in context: A264864 A297472 A299129 * A256153 A238196 A352347

Adjacent sequences: A255674 A255675 A255676 * A255678 A255679 A255680

Keyword

nonn

Author

Zhi-Wei Sun, Jul 10 2015

Status

approved