cp's OEIS Frontend

A292509 Primes of the form k^2 + 23*k + 23.

%I A292509 #54 Apr 15 2025 11:02:12
%S A292509 23,47,73,101,131,163,197,233,271,311,353,397,443,491,541,593,647,761,
%T A292509 821,883,947,1013,1151,1223,1297,1373,1451,1531,1613,1697,1783,1871,
%U A292509 2053,2243,2341,2441,2543,2647,2753,2861,2971,3083,3313,3673,3797,3923,4051,4447
%N A292509 Primes of the form k^2 + 23*k + 23.
%C A292509 The first 17 primes correspond to n from 0 to 16, which makes n^2 + 23n + 23 a prime-generating polynomial (see the link). This is a monic polynomial of the form n^2 + pn + p, where p is prime. Among the first 10^8 primes, only two more besides 23 give rise to prime-generating polynomials of this form. They are 8693 and 50983511 and they generate only 11 primes for n = 0 to 10.
%H A292509 Robert Israel, <a href="/A292509/b292509.txt">Table of n, a(n) for n = 1..10000</a>
%H A292509 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%e A292509 For n = 1, we have 1^2 + 23 * 1 + 23 = 47, which is prime, so 47 is in the sequence.
%e A292509 For n = 2, we have 2^2 + 23 * 2 + 23 = 4 + 46 + 23 = 73, which is prime, so 73 is in the sequence.
%e A292509 Contrast to n = 17, which gives us 17^2 + 23 * 17 + 23 = 289 + 391 + 23 = 703 = 19 * 37, so 703 is not in the sequence.
%p A292509 select(isprime, [seq(x^2+23*x+23, x=0..1000)]); # _Robert Israel_, Sep 18 2017
%t A292509 Select[Range[0, 100]//#^2 + 23# + 23 &, PrimeQ]
%o A292509 (PARI) for(n=0, 100, isprime(n^2+23*n+23)&&print1(n^2+23*n+23 ","))
%o A292509 (Magma) [a: n in [0..100] | IsPrime(a) where a is n^2+23*n+23 ]; // _Vincenzo Librandi_, Sep 23 2017
%Y A292509 Cf. A005846, A300473.
%K A292509 nonn
%O A292509 1,1
%A A292509 _Waldemar Puszkarz_, Sep 17 2017