A056899 Primes of the form k^2 + 2.

2, 3, 11, 83, 227, 443, 1091, 1523, 2027, 3251, 6563, 9803, 11027, 12323, 13691, 15131, 21611, 29243, 47963, 50627, 56171, 59051, 62003, 65027, 74531, 88211, 91811, 95483, 103043, 119027, 123203, 131771, 136163, 140627, 149771, 173891

Offset

1,1

Comments

Also, primes of the form k^2 - 2k + 3.

Note that all terms after the first two are equal to 11 modulo 72 and that (a(n)-11)/72 is a triangular number, since they have to be 2 more than the square of an odd multiple of 3 to be prime, and if k = 6*m+3 then a(n) = k^2 + 2 = 72*m*(m+1)/2 + 11.

The quotient cycle length is 2 in the continued fraction expansion of sqrt(p) for these primes. E.g.: cfrac(sqrt(6563),6) = 81+1/(81+1/(162+1/(81+1/(162+1/(81+1/(162+`...`)))))). - Labos Elemer, Feb 22 2001

Primes in A059100; except for a(2)=3 a subsequence of A007491 and congruent to 2 modulo 9. For n>2, a(n)=11 (mod 72). - M. F. Hasler, Apr 05 2009

References

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.

Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Near-Square Prime

Formula

For n>1, a(n) = 72*A000217(A056900(n-2))+11

a(n) = A067201(n)^2 + 2. - M. F. Hasler, Apr 05 2009

Maple

select(isprime, [seq(t^2+2, t = 0..1000)]); # Robert Israel, Sep 03 2015

Mathematica

Select[ Range[0, 500]^2 + 2, PrimeQ] (* Robert G. Wilson v, Sep 03 2015 *)

Prog

(Magma) [n: n in PrimesUpTo(175000) | IsSquare(n-2)];  // Bruno Berselli, Apr 05 2011
(Magma) [ a: n in [0..450] | IsPrime(a) where a is n^2 +2 ]; // Vincenzo Librandi, Apr 06 2011
(PARI) print1("2, 3"); forstep(n=3, 1e4, 6, if(isprime(t=n^2+2), print1(", "t))) \\ Charles R Greathouse IV, Jul 19 2011

Crossrefs

Intersection of A146327 and A000040; intersection of A059100 and A000040.

Cf. A002496.

Sequence in context: A290512 A042165 A089921 * A347402 A117699 A065378

Adjacent sequences: A056896 A056897 A056898 * A056900 A056901 A056902

Keyword

nonn

Author

Henry Bottomley, Jul 05 2000

Status

approved