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

Author

Henry Bottomley, Jul 05 2000

Keywords

• nonn

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

Crossrefs

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

Cf. A002496.

Programs

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

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 *)

PARI

print1("2, 3");forstep(n=3,1e4,6,if(isprime(t=n^2+2),print1(", "t))) \\ Charles R Greathouse IV, Jul 19 2011

Formula

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

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