

A048059


Primes of the form k^2 + k + 11.


17



11, 13, 17, 23, 31, 41, 53, 67, 83, 101, 167, 193, 251, 283, 317, 353, 431, 563, 661, 823, 881, 941, 1201, 1493, 1571, 1733, 2081, 2267, 2663, 2767, 3203, 3433, 3671, 3793, 3917, 4567, 4703, 5413, 5711, 6173, 6491, 6653, 6983, 7151, 7321
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OFFSET

1,1


COMMENTS

The polynomial P(n) := n^2 + n + 11 takes distinct prime values for the 10 consecutive integers n = 0 to 9. It follows that the polynomial P(n10) = (n  10)^2 + (n  10) + 11 takes prime values for the 20 consecutive integers n = 0 to 19, consisting of the 10 primes above each taken twice. We note two consequences of this fact.
1) The polynomial P(2*n10) = 4*n^2  38*n + 101 also takes prime values for the 10 consecutive integers n = 0 to 9.
2)The polynomial P(3*n10) = 9*n^2  57*n + 101 takes prime values for the 7 consecutive integers n = 0 to 6 (= floor(19/3)). In addition, calculation shows that P(3*n10) also takes prime values for n from 3 to 1. Equivalently put, the polynomial P(3*n19) = 9*n^2  111*n + 353 takes prime values for the 10 consecutive integers n = 0 to 9. Cf. A007635 and A005846. (End)


LINKS



MATHEMATICA



PROG

(Magma) [ a: n in [0..200]  IsPrime(a) where a is n^2+n+11 ]; // Vincenzo Librandi, Dec 07 2011


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



