A028871 Primes of the form k^2 - 2.

2, 7, 23, 47, 79, 167, 223, 359, 439, 727, 839, 1087, 1223, 1367, 1847, 2207, 2399, 3023, 3719, 3967, 4759, 5039, 5623, 5927, 7919, 8647, 10607, 11447, 13687, 14159, 14639, 16127, 17159, 18223, 19319, 21023, 24023, 25919, 28559, 29927

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• easy

Comments

Except for the initial term, primes equal to the product of two consecutive even numbers minus 1. - Giovanni Teofilatto, Sep 24 2004

With exception of the first term 2, primes p such that continued fraction of (1+sqrt(p))/2 have period 4. - Artur Jasinski, Feb 03 2010

Subsequence of A094786. First primes q that are in A094786 but not here are 241, 3373, 6857, 19681, 29789, 50651, 300761, 371291, ...; q+2 are perfect powers m^k with odd k>1. - Zak Seidov, Dec 09 2014

References

• D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, p. 31.

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• P. De Geest, Palindromic Quasipronics of the form n(n+x)
• Eric Weisstein's World of Mathematics, Near-Square Prime
• Wikipedia, Bunyakovsky conjecture

Crossrefs

Cf. A028870, A089623, A010051, A094786; subsequence of A008865.

Programs

Haskell

a028871 n = a028871_list !! (n-1)
a028871_list = filter ((== 1) . a010051') a008865_list
-- Reinhard Zumkeller, May 06 2013

Magma

[p: p in PrimesUpTo(100000)| IsSquare(p+2)]; // Vincenzo Librandi, Jun 19 2014

Maple

select(isprime, [2,seq((2*n+1)^2-2, n=1..1000)]); # Robert Israel, Dec 09 2014

Mathematica

lst={};Do[s=n^2;If[PrimeQ[p=s-2], AppendTo[lst, p]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 26 2008 *)
aa = {}; Do[If[4 == Length[ContinuedFraction[(1 + Sqrt[Prime[m]])/2][[2]]], AppendTo[aa, Prime[m]]], {m, 1, 1000}]; aa (* Artur Jasinski, Feb 03 2010 *)
Select[Table[n^2 - 2, {n, 400}], PrimeQ] (* Vincenzo Librandi, Jun 19 2014 *)

PARI

list(lim)=select(n->isprime(n),vector(sqrtint(floor(lim)+2),k,k^2-2)) \\ Charles R Greathouse IV, Jul 25 2011

Formula

a(n) = A028870(n)^2 -2. - R. J. Mathar, Dec 12 2023