A048988 - OEIS

%I #49 Aug 25 2024 13:34:25

%S 59,67,83,107,139,179,227,283,347,419,499,587,683,787,1019,1283,1427,

%T 1579,1907,2083,2267,2459,2659,3083,3307,3539,3779,4027,4283,4547,

%U 5099,5387,5683,5987,6299,6619,6947,7283,8707,9467,9859,10259,10667,11083

%N Primes of the form 4*k^2 + 4*k + 59.

%C From _Peter Bala_, Apr 18 2018: (Start)

%C Let P(n) = 4*n^2 + 4*n + 59. The polynomial 1/2*P(n-1/2) = 2*n^2 + 29 has prime values for n from 0 to 28. See A007641. Also P(n-14) = 4*n^2 - 108*n + 787 is prime for the 28 consecutive values of n from 0 to 27.

%C The sequence of 28 values of the polynomial 4*P((n-2)/4) = n^2 + 232 for n from -1 to 26 is [233, 2^3*29, 233, 2^2*59, 241, 2^3*31, 257, 2^2*67, 281, 2^3*37, 313, 2^2*83, 353, 2^3*47, 401, 2^2*107, 457, 2^3*61, 521, 2^2*139, 593, 2^3*79, 673, 2^2*179, 761, 2^3*101, 857, 2^2*227], and consists of 7 groups of 4 numbers of the form p_1, 2^3*p_2, p_3, 2^2*p_4, where the p's are prime numbers. (End)

%H Seiichi Manyama, <a href="/A048988/b048988.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial.</a>

%o (Magma) [ a: n in [0..250] | IsPrime(a) where a is 4*n^2 +4*n + 59]; // _Vincenzo Librandi_, Nov 19 2010

%o (PARI) lista(nn) = for(k=0, nn, if(isprime(p=4*k^2+4*k+59), print1(p, ", "))); \\ _Altug Alkan_, Apr 18 2018

%Y Cf. A005846, A007641, A271980.

%K nonn,easy,less

%O 1,1

%A _G. L. Honaker, Jr._