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A371896 a(n) is the length of the uninterrupted sequence of primes generated by the polynomial f(x) = x^2 + x + p for x=0,1,..., where p=A001359(n).

%I A371896 #46 Jun 02 2024 14:40:20
%S A371896 2,4,10,16,2,40,2,2,4,3,2,2,2,3,2,4,2,2,2,3,5,2,2,3,2,2,2,2,5,2,2,3,2,
%T A371896 3,3,2,2,2,2,4,2,2,7,2,3,2,5,2,4,4,6,2,2,2,2,2,3,2,2,2,3,2,3,2,2,2,2,
%U A371896 3,3,2,2,2,2,2,3,5,2,2,2,2,2,2,2,2,3,2
%N A371896 a(n) is the length of the uninterrupted sequence of primes generated by the polynomial f(x) = x^2 + x + p for x=0,1,..., where p=A001359(n).
%C A371896 p=A001359(n) is the smaller prime of a twin prime pair so that f(0) = p and f(1) = p+2 are both primes so a(n) >= 2 and this sequence is the terms >= 2 in A208936.
%D A371896 L. Euler, Nouveaux Mémoires de l'Académie royale des Sciences, 1772, p. 36.
%H A371896 Peter Rowlett, <a href="/A371896/b371896.txt">Table of n, a(n) for n = 1..10000</a>
%e A371896 For n=6, p = A001359(n) = 41 and f(x) = x^2 + x + 41 is Euler's polynomial which generates primes f(x) for x=0,1,2,...,39, which is 40 terms so a(6) = 40 (cf. A202018).
%t A371896 A001359[1] = 3;
%t A371896 A001359[n_] := A001359[n] = (p = NextPrime[A001359[n - 1]];
%t A371896   While[!PrimeQ[p + 2], p = NextPrime[p]]; p)
%t A371896 A371896[n_] := With[{p = A001359@n}, k = 1;
%t A371896   While[PrimeQ[k^2 + k + p], k++]; k]
%t A371896 (* _Leo C. Stein_, May 09 2024 *)
%o A371896 (C++)
%o A371896 #include <iostream>
%o A371896 using namespace std;
%o A371896 bool isPrime(int number){
%o A371896     if(number < 2) return false;
%o A371896     if(number == 2) return true;
%o A371896     if(number % 2 == 0) return false;
%o A371896     for(int i=3; (i*i)<=number; i+=2){
%o A371896         if(number % i == 0) return false;
%o A371896     }
%o A371896     return true;
%o A371896 }
%o A371896 int main() {
%o A371896     int x;
%o A371896     for (int p=2; p<100000000; p++) {
%o A371896         if (isPrime(p)) {
%o A371896             x=1;
%o A371896             while (isPrime(x*x+x+p)) {
%o A371896                 x++;
%o A371896             }
%o A371896             if (x>1) {
%o A371896                 cout << x << ", ";
%o A371896             }
%o A371896         }
%o A371896     }
%o A371896     return 0;
%o A371896 }
%Y A371896 Cf. A001359, A202018, A208936.
%K A371896 nonn
%O A371896 1,1
%A A371896 _Peter Rowlett_, Apr 11 2024