A110013 -id:A110013 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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.

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

Cf. A010051.

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

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

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

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

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

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

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

Extended by Ray Chandler, Sep 07 2005

A002328 Numbers k such that k^2 - k - 1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 10, 11, 12, 14, 16, 17, 20, 21, 22, 25, 27, 29, 31, 32, 36, 39, 40, 42, 45, 46, 47, 49, 51, 54, 55, 56, 57, 60, 61, 65, 66, 67, 69, 71, 77, 84, 86, 87, 90, 94, 95, 97, 101, 102, 104, 115, 116, 121, 126, 127, 131, 132, 135, 139, 141, 142, 145, 146, 149

Offset: 1

N. J. A. Sloane

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).

Carmine Suriano, Table of n, a(n) for n = 1..60000

Cf. A002327, A088502, A110013.

[k: k in [1..300]|IsPrime(k^2-k-1)]; // Vincenzo Librandi, Nov 21 2010

Select[Range[10^3], PrimeQ[#^2 - # - 1] &] (* Vincenzo Librandi, Mar 20 2014 *)

is(k)=isprime(k^2 - k - 1) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = (A088502(n)+1)/2. - Ray Chandler

a(n) = A094210(n) + 2. - R. J. Mathar, Aug 08 2012

Extended by Ray Chandler, Sep 07 2005

A088502 Numbers n such that (n^2 - 5)/4 is prime.

Original entry on oeis.org

5, 7, 9, 11, 13, 17, 19, 21, 23, 27, 31, 33, 39, 41, 43, 49, 53, 57, 61, 63, 71, 77, 79, 83, 89, 91, 93, 97, 101, 107, 109, 111, 113, 119, 121, 129, 131, 133, 137, 141, 153, 167, 171, 173, 179, 187, 189, 193, 201, 203, 207, 229, 231, 241, 251, 253, 261, 263, 269

Offset: 1

Pierre CAMI, Nov 13 2003

Under Bunyakovsky's conjecture this sequence is infinite. - Charles R Greathouse IV, Dec 28 2011

			(23*23 - 5)/4 = 131, 131 is prime, 23 is the 9th n of the sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002327, A002328, A110013.

[n: n in [1..300 by 2] | IsPrime((n^2-5) div 4)]; // Vincenzo Librandi, Oct 06 2012

Select[Range[500], PrimeQ[(#^2 - 5)/4] &] (* Vincenzo Librandi, Oct 06 2012 *)

for(k=2,1e3,if(isprime(k^2+k-1),print1(2*k+1", "))) \\ Charles R Greathouse IV, Dec 28 2011

a(n) = 2*A002328(n) - 1 = Sqrt(A110013(n)). - Ray Chandler, Sep 07 2005

A110484 Squares of the form p*q + p + q + 2, where p and q are primes.

Original entry on oeis.org

25, 49, 81, 121, 169, 289, 361, 441, 529, 625, 729, 841, 961, 1089, 1225, 1369, 1521, 1681, 1849, 2209, 2401, 2809, 3025, 3249, 3481, 3721, 3969, 4225, 4489, 5041, 5329, 5929, 6241, 6889, 7225, 7921, 8281, 8649, 9025, 9409, 10201, 10609, 11449, 11881

Offset: 1

Giovanni Teofilatto, Sep 10 2005

nonn

Includes A108604 squares of greater twin primes and A110013 squares of form 4p+5 where p is prime (q=3).

Cf. A108604, A110013, A110487.

Edited by Ray Chandler, Sep 13 2005

cp's OEIS Frontend

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

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A002328 Numbers k such that k^2 - k - 1 is prime.

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A088502 Numbers n such that (n^2 - 5)/4 is prime.

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Magma

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A110484 Squares of the form p*q + p + q + 2, where p and q are primes.

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Crossrefs

Extensions