cp's OEIS Frontend

A300473 Numbers k with the property that k^2 + 21k + 1 is prime.

%I A300473 #39 Feb 16 2025 08:33:53
%S A300473 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19,20,21,22,23,25,26,27,28,
%T A300473 29,30,31,32,33,34,36,38,39,40,41,42,43,44,45,46,48,51,52,53,54,57,58,
%U A300473 59,60,61,62,64,65,66,67,68,69,73,77,78,79,80,81,82,84,85,86,87,89,91,97,100
%N A300473 Numbers k with the property that k^2 + 21k + 1 is prime.
%C A300473 The quadratic polynomial p(k) = k^2 + 21*k + 1 is not a prime-generating polynomial in the sense of Eric Weisstein's World of Mathematics (see link) because p(0) is not prime.
%C A300473 However p(k) is prime for the first 17 positive integral values of k and among polynomials of the form k^2 + j*k + 1, the present polynomial (j = 21) generates more primes than any polynomial of that form for any positive integral j < 231, at least for positive integers, k, in the range 0 < k < 10^6.
%H A300473 Seiichi Manyama, <a href="/A300473/b300473.txt">Table of n, a(n) for n = 1..10000</a>
%H A300473 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>
%e A300473 17 is in the sequence because 17^2 + 21 * 17 + 1 = 647 is prime.
%e A300473 18 is not in the sequence because 18^2 + 21 * 18 + 1 = 703 = 19 * 37.
%p A300473 select(k-> isprime(k^2+21*k+1), [$1..100])
%t A300473 Select[Range[100], PrimeQ[#^2 + 21# + 1] &] (* _Alonso del Arte_, Mar 06 2018 *)
%o A300473 (PARI) isok(k) = isprime(k^2+21*k+1); \\ _Altug Alkan_, Mar 07 2018
%Y A300473 Cf. A292509, A050267, A050268, A005846, A007635, A007641, A048988.
%K A300473 nonn
%O A300473 1,2
%A A300473 _James R. Buddenhagen_, Mar 06 2018