A002327 Primes of the form k^2 - k - 1.

5, 11, 19, 29, 41, 71, 89, 109, 131, 181, 239, 271, 379, 419, 461, 599, 701, 811, 929, 991, 1259, 1481, 1559, 1721, 1979, 2069, 2161, 2351, 2549, 2861, 2969, 3079, 3191, 3539, 3659, 4159, 4289, 4421, 4691, 4969, 5851, 6971, 7309, 7481, 8009, 8741, 8929

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also primes of form x*y + x + y or x*y - x - y, where x and y are two successive numbers. - Giovanni Teofilatto, May 12 2004

Equivalently primes p such that 4p+5 is square. - Giovanni Teofilatto, Sep 03 2005

Arithmetic numbers which are triangular, A003601(p)=A000217(k), p prime. sigma_1(p)/sigma_0(p) = k*(k+1)/2; sigma_r(p) divisor function, p prime, k integer. - Ctibor O. Zizka, Jul 14 2008

Also primes of the form k^2 + 3k + 1 (primes in A028387). - Zak Seidov, Apr 13 2014

Also primes p such that the sum of divisors (A000203) of p is oblong (A002378). - Michel Marcus, Jan 09 2015

References

• D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 46.
• L. Poletti, Tavole di Numeri Primi Entro Limiti Diversi e Tavole Affini, Milan, 1920, p. 249.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi and Pierre CAMI, Table of n, a(n) for n = 1..10000 (Vincenzo Librandi to n=1000)
• Marie Euler and Christophe Petit, Expanding the use of quasi-subfield polynomials, arXiv:1909.11326 [cs.CR], 2019.

Crossrefs

Cf. A002328, A088502, A110013, A003601, A000217, A028387.

Cf. A010051.

Programs

Haskell

a002327 n = a002327_list !! (n-1)
a002327_list = filter ((== 1) .  a010051') a028387_list
-- Reinhard Zumkeller, Jul 17 2014

Magma

[ a: n in [0..150] | IsPrime(a) where a is n^2 - n - 1 ]; // Vincenzo Librandi, Aug 01 2011

Maple

A002327:=n->`if`(isprime(n^2-n-1), n^2-n-1, NULL): seq(A002327(n), n=1..100); # Wesley Ivan Hurt, Aug 09 2014

Mathematica

Select[Table[n^2-n-1,{n,100}],PrimeQ] (* Harvey P. Dale, May 03 2011 *)

PARI

for(n=2,1e3,if(isprime(k=n^2-n-1),print1(k", "))) \\ Charles R Greathouse IV, Jul 31 2011

PARI

list(lim)=my(v=List(),p); forstep(n=5,sqrtint(4*lim+5),2, if(isprime(p=(n^2-5)/4), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Oct 10 2023

Formula

a(n) = A002328(n)^2 - A002328(n) - 1 = (A110013(n) - 5)/4. - Ray Chandler, Sep 07 2005

a(n) >> n^2 log n by Brun's sieve. - Charles R Greathouse IV, Oct 10 2023

Extensions

Extended by Ray Chandler, Sep 07 2005