A090106 Values of k such that {P(k), P(k+1), ..., P(k+12)} are all prime numbers, where P(k) = k^2 + k + 41.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 219

Offset: 1

Author

Labos Elemer, Dec 22 2003

Keywords

• nonn
• more

Comments

a(n) is the first argument providing 13 "polynomially consecutive" primes with respect to the polynomial x^2 + x + 41.

a(29) > 5*10^9, if it exists. - Amiram Eldar, Sep 27 2024

Examples

			k = 219: {P(219), ..., P(231)} = {48221, ..., 53633}, i.e., 13 consecutive integer values substituted to P(x) = x^2 + x + 41 polynomial, all provide primes. The "classical case" includes one single 41-chain of PC-primes, see A055561.
		

Crossrefs

Cf. A055561, A090101, A090102, A090562, A090563.

Programs

Mathematica

Position[Times @@@ Partition[Table[Boole@PrimeQ[k^2 + k + 41], {k, 1, 1000}], 13, 1], 1] // Flatten (* Amiram Eldar, Sep 27 2024 *)

PARI

isp(x) = isprime(x^2 + x + 41);
lista(kmax) = {my(v = vector(13, k, isp(k))); for(k = 14, kmax, if(vecprod(v) == 1, print1(k - 13, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024

Extensions

2 wrong terms removed by Amiram Eldar, Sep 27 2024