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A261354 Primes p such that prime(p)^2 - 2 = prime(q) for some prime q. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)