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A090107 Values of k such that {P(k), P(k+1), ..., P(k+9)} are all prime numbers, where P(k) = k^2 - 79*k + 1601.

%I A090107 #13 Sep 27 2024 08:35:45
%S A090107 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,
%T A090107 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,
%U A090107 50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,106
%N A090107 Values of k such that {P(k), P(k+1), ..., P(k+9)} are all prime numbers, where P(k) = k^2 - 79*k + 1601.
%C A090107 a(n) is the first argument providing 10 "polynomially consecutive" primes with respect to the polynomial x^2 - 79*x + 1601 described by Escott in 1899.
%H A090107 Amiram Eldar, <a href="/A090107/b090107.txt">Table of n, a(n) for n = 1..145</a>
%H A090107 E. B. Escott, <a href="https://archive.org/details/lintermdiairede03lemogoog/page/n336/mode/2up">Réponse 1133, Formule d'Euler x^2 + x + 41 et formules analogues</a>, L'Intermédiaire des mathématiciens, Vol. 6 (1899), pp. 10-11.
%e A090107 k = 106 provides a chain of 10 "polynomially consecutive" primes as follows: {4463, 4597, 4733, 4871, 5011, 5153, 5297, 5443, 5591, 5741}.
%t A090107 Position[Times @@@ Partition[Table[Boole@PrimeQ[k^2 - 79*k + 1601], {k, 1, 1000}], 10, 1], 1] // Flatten (* _Amiram Eldar_, Sep 27 2024 *)
%o A090107 (PARI) isp(x) = isprime(x^2 - 79*x + 1601);
%o A090107 lista(kmax) = {my(v = vector(10, k, isp(k))); for(k = 11, kmax, if(vecprod(v) == 1, print1(k - 10, ", ")); v = concat(vecextract(v, "^1"), isp(k)));} \\ _Amiram Eldar_, Sep 27 2024
%Y A090107 Cf. A055561, A090101, A090102, A090562, A090563.
%K A090107 nonn
%O A090107 1,2
%A A090107 _Labos Elemer_, Dec 29 2003
%E A090107 Data corrected by _Amiram Eldar_, Sep 27 2024