A090108 Values of k such that {P(k), P(k+1), ..., P(k+8)} are all prime numbers, whereP(k) = k^2 - 79*k + 1601.

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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 106

Offset: 1

Author

Labos Elemer, Dec 29 2003

Keywords

• nonn

Comments

a(n) is the first argument providing 9 "polynomially consecutive" primes with respect to the polynomial x^2 - 79*x + 1601 described by Escott in 1899.

Examples

			k = 263 provides a chain of 9 "polynomially consecutive" primes as follows: {49993, 50441, 50891, 51343, 51797, 52253, 52711, 53171, 53633}.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..535
• E. B. Escott, Réponse 1133, Formule d'Euler x^2 + x + 41 et formules analogues, L'Intermédiaire des mathématiciens, Vol. 6 (1899), pp. 10-11.

Crossrefs

Cf. A055561, A090101, A090102, A090107, A090562, A090563.

Programs

Mathematica

Position[Times @@@ Partition[Table[Boole@PrimeQ[k^2 - 79*k + 1601], {k, 1, 1000}], 9, 1], 1] // Flatten (* Amiram Eldar, Sep 27 2024 *)

PARI

isp(x) = isprime(x^2 - 79*x + 1601);
lista(kmax) = {my(v = vector(9, k, isp(k))); for(k = 10, kmax, if(vecprod(v) == 1, print1(k - 9, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ Amiram Eldar, Sep 27 2024