A260219 Least prime p such that pi(p*n)^2 + 1 = prime(q*n) for some prime q.

3, 47, 229, 379, 11, 2687, 1181, 24547, 5, 509, 5, 16619, 877, 22543, 3, 9067, 60337, 10667, 997, 24061, 18329, 56099, 1787, 58757, 108883, 416881, 157141, 11003, 14939, 113167, 101957, 77969, 613, 27947, 52153, 158551, 6197, 36607, 25237, 27179, 330689, 203617, 77419, 708269, 87649, 340381, 267601, 67153, 123377, 21617

Offset: 1

Author

Zhi-Wei Sun, Jul 19 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. In general, if a,b,c are integers with a > 0 and gcd(a,b,c) = 1, and a+b or c is odd, and b^2 - 4*a*c is not a square, then there are primes p and q such that a*pi(p*n)^2 + b*pi(p*n) + c = prime(q*n).

Examples

			a(1) = 3 since pi(3*1)^2 + 1 = 5 = prime(3*1) with 3 prime.
a(8) = 24547 since pi(24547*8)^2 + 1 = 17686^2 + 1 = 312794597 = prime(2113417*8) with 24547 and 2113417 both prime.
		

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..72
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000720, A002496, A259731, A260120, A260197.

Programs

Mathematica

PQ[n_, p_] := PrimeQ[p] && PrimeQ[PrimePi[p]/n]; Do[k = 0; Label[bb]; k = k + 1; If[PQ[n, PrimePi[Prime[k] * n]^2 + 1], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]]; Continue,{n, 50}]

PARI

first(m)={my(v=vector(m),p,q,n);for(n=1,m,p=0;while(1,p++;q=1;while(primepi(prime(p)*n)^2 +1 >= prime(prime(q)*n), if(primepi(prime(p)*n)^2 +1 == prime(prime(q)*n),v[n]=prime(p);break(2),q++;))));v;} /* Anders Hellström, Jul 19 2015 */