A007635 Primes of form n^2 + n + 17. (Formerly M5069)

17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 359, 397, 479, 523, 569, 617, 719, 773, 829, 887, 947, 1009, 1277, 1423, 1499, 1657, 1823, 1997, 2087, 2179, 2273, 2467, 2879, 3209, 3323, 3557, 3677, 3923, 4049, 4177, 4987, 5273

Offset

1,1

Comments

a(n) = A117530(7,n) for n <= 7: a(1) = A117530(7,1) = A014556(5) = 17, A117531(7) = 7. - Reinhard Zumkeller, Mar 26 2006

Note that the gaps between terms increases by 2*k from k = 1 to 15: 19 - 17 = 2, 23 - 19 = 4, 29 - 23 = 6 and so on until 257 - 227 = 30 then fails at 289 - 257 = 32 since 289 = 17^2. - J. M. Bergot, Mar 18 2017

From Peter Bala, Apr 15 2018: (Start)

The polynomial P(n):= n^2 + n + 17 takes distinct prime values for the 16 consecutive integers n = 0 to 15. It follows that the polynomial P(n - 16) takes prime values for the 32 consecutive integers n = 0 to 31, consisting of the 16 primes above each taken twice. We note two consequences of this fact.

1) The polynomial P(2*n - 16) = 4*n^2 - 62*n + 257 also takes prime values for the 16 consecutive integers n = 0 to 15.

2)The polynomial P(3*n - 16) = 9*n^2 - 93*n + 257 takes prime values for the 11 consecutive integers n = 0 to 10 ( = floor(31/3)). In addition, calculation shows that P(3*n-16) also takes prime values for n from -5 to -1. Equivalently put, the polynomial P(3*n-31) = 9*n^2 - 183*n + 947 takes prime values for the 16 consecutive integers n = 0 to 15. Cf. A005846 and A048059. (End)

The primes in this sequence are not primes in the ring of integers of Q(sqrt(-67)). If p = n^2 + n + 17, then ((2n + 1)/2 - sqrt(-67)/2)((2n + 1)/2 + sqrt(-67)/2) = p. For example, 3^2 + 3 + 17 = 29 and (7/2 - sqrt(-67)/2)(7/2 + sqrt(-67)/2) = 29 also. - Alonso del Arte, Nov 27 2019

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 96.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime-generating Polynomial.

Formula

a(n) = A028823(n)^2 + A028823(n) + 17. - Seiichi Manyama, Mar 19 2017

Mathematica

Select[Table[n^2 + n + 17, {n, 0, 99}], PrimeQ] (* Alonso del Arte, Nov 27 2019 *)

Prog

(Magma) [a: n in [0..250]|IsPrime(a) where a is n^2+n+17] // Vincenzo Librandi, Dec 23 2010
(PARI) select(isprime, vector(100, n, n^2+n+17)) \\ Charles R Greathouse IV, Jul 12 2016
(Python)
from sympy import isprime
it = (n**2 + n + 17 for n in range(250))
print([p for p in it if isprime(p)]) # Indranil Ghosh, Mar 18 2017

Crossrefs

Cf. A005846, A028823, A048059, A160548.

Sequence in context: A106933 A191041 A106932 * A140947 A205700 A228070

Adjacent sequences: A007632 A007633 A007634 * A007636 A007637 A007638

Keyword

nonn,easy

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Status

approved