A255679 Least number k such that pi(k*n)^2 = pi(i*n)^2 + pi(j*n)^2 for some 0 < i < j < k.

4, 6, 4, 12, 6, 17, 17, 6, 11, 6, 13, 14, 4, 10, 31, 35, 10, 8, 20, 20, 70, 20, 34, 23, 47, 17, 44, 25, 5, 46, 4, 26, 53, 13, 29, 33, 35, 45, 65, 10, 18, 64, 117, 21, 77, 19, 71, 101, 44, 80, 28, 93, 36, 183, 100, 98, 55, 4, 53, 81, 31, 120, 58, 40, 53, 56, 34, 49, 181, 218

Offset: 1

Author

Zhi-Wei Sun, Jul 11 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. In other words, for each n = 1,2,3,... the sequence pi(k*n) (k = 1,2,3,...) contains a Pythagorean triple.

Examples

			a(1) = 4 since pi(4*1)^2 = 2^2 = pi(1*1)^2 + pi(3*1)^2.
a(21) = 70 since pi(70*21)^2 = pi(1470)^2 = 232^2 = 160^2 + 168^2 = pi(45*21)^2 + pi(48*21)^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 = 1..240
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000290, A000720, A255677.

Programs

Mathematica

Do[k=0;Label[bb];k=k+1;Do[If[PrimePi[k*n]^2==PrimePi[i*n]^2+PrimePi[j*n]^2,Goto[aa],Goto[cc]];Label[cc];Continue,{j,1,k-1},{i,1,j-1}];Goto[bb];
Label[aa];Print[n," ",k];Continue,{n,1,70}]

PARI

is_ok(n,k)={ r=0; for(i=1,k-1, for(j=i+1, k-1, if(primepi(k*n)^2 == primepi(i*n)^2 + primepi(j*n)^2,r=1;break(2)) )); return(r);}
a(n)={ k=0; while(!is_ok(n,k), k++); return(k);}
main(size)={ v=vector(size); i=1; m=1; while(i<=size, v[i]=a(i);i++); return(v);} /* Anders Hellström, Jul 11 2015 */