A261354 Primes p such that prime(p)^2 - 2 = prime(q) for some prime q.

31, 191, 541, 809, 1153, 1301, 2221, 3037, 3847, 4049, 4159, 5441, 8243, 10177, 12277, 13681, 14783, 15619, 17903, 19463, 20897, 22697, 24517, 25163, 25847, 25849, 26633, 26647, 27329, 27407, 28051, 32653, 35059, 35747, 36341, 36527, 37369, 37811, 38609, 40949, 42737, 46679, 51061, 51607, 54443, 54679, 56113, 57637, 60887, 61493

Offset

1,1

Comments

Conjecture: The sequence has infinitely many terms. In general, for any integers a,b,c with a>0 and gcd(a,b,c)=1, if b^2-4*a*c is not a square, a+b+c is odd, and gcd(b,a+c) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2+b*prime(p)+c = prime(q).

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..100

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

Example

a(1) = 31 since 31 is a prime, and prime(31)^2-2 = 127^2-2 = 16127 = prime(1877) with 1877 prime.

Mathematica

PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
f[k_]:=Prime[Prime[k]]^2-2
n=0; Do[If[PQ[f[k]], n=n+1; Print[n, " ", Prime[k]]], {k, 1, 6200}]

Crossrefs

Cf. A000040, A049002, A062326, A237413, A260120, A261281, A261352, A261361.

Sequence in context: A042878 A165130 A229145 * A023292 A100689 A055816

Adjacent sequences: A261351 A261352 A261353 * A261355 A261356 A261357

Keyword

nonn

Author

Zhi-Wei Sun, Aug 15 2015

Status

approved