A048988 Primes of the form 4*k^2 + 4*k + 59.

59, 67, 83, 107, 139, 179, 227, 283, 347, 419, 499, 587, 683, 787, 1019, 1283, 1427, 1579, 1907, 2083, 2267, 2459, 2659, 3083, 3307, 3539, 3779, 4027, 4283, 4547, 5099, 5387, 5683, 5987, 6299, 6619, 6947, 7283, 8707, 9467, 9859, 10259, 10667, 11083

Offset: 1

Author

G. L. Honaker, Jr.

Keywords

• nonn
• easy
• less

Comments

From Peter Bala, Apr 18 2018: (Start)

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.

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)

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Crossrefs

Cf. A005846, A007641, A271980.

Programs

Magma

[ a: n in [0..250] | IsPrime(a) where a is 4*n^2 +4*n + 59]; // Vincenzo Librandi, Nov 19 2010

Maple

select(isprime, [4*k*(k+1)+59$k=0..100])[];  # Alois P. Heinz, Apr 16 2025

Mathematica

Select[(4 #^2 + 4 # + 59) & /@ Range[0, 100], PrimeQ] (* Robert Price, Apr 16 2025 *)

PARI

lista(nn) = for(k=0, nn, if(isprime(p=4*k^2+4*k+59), print1(p, ", "))); \\ Altug Alkan, Apr 18 2018