login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002327 Primes of the form k^2 - k - 1.
(Formerly M3810 N1558)
41

%I M3810 N1558 #91 Oct 10 2023 23:01:46

%S 5,11,19,29,41,71,89,109,131,181,239,271,379,419,461,599,701,811,929,

%T 991,1259,1481,1559,1721,1979,2069,2161,2351,2549,2861,2969,3079,3191,

%U 3539,3659,4159,4289,4421,4691,4969,5851,6971,7309,7481,8009,8741,8929

%N Primes of the form k^2 - k - 1.

%C 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

%C Equivalently primes p such that 4p+5 is square. - _Giovanni Teofilatto_, Sep 03 2005

%C 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

%C Also primes of the form k^2 + 3k + 1 (primes in A028387). - _Zak Seidov_, Apr 13 2014

%C Also primes p such that the sum of divisors (A000203) of p is oblong (A002378). - _Michel Marcus_, Jan 09 2015

%D D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 46.

%D L. Poletti, Tavole di Numeri Primi Entro Limiti Diversi e Tavole Affini, Milan, 1920, p. 249.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi and Pierre CAMI, <a href="/A002327/b002327.txt">Table of n, a(n) for n = 1..10000</a> (Vincenzo Librandi to n=1000)

%H Marie Euler and Christophe Petit, <a href="https://arxiv.org/abs/1909.11326">Expanding the use of quasi-subfield polynomials</a>, arXiv:1909.11326 [cs.CR], 2019.

%F a(n) = A002328(n)^2 - A002328(n) - 1 = (A110013(n) - 5)/4. - _Ray Chandler_, Sep 07 2005

%F a(n) >> n^2 log n by Brun's sieve. - _Charles R Greathouse IV_, Oct 10 2023

%p 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

%t Select[Table[n^2-n-1,{n,100}],PrimeQ] (* _Harvey P. Dale_, May 03 2011 *)

%o (PARI) for(n=2,1e3,if(isprime(k=n^2-n-1),print1(k", "))) \\ _Charles R Greathouse IV_, Jul 31 2011

%o (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

%o (Magma) [ a: n in [0..150] | IsPrime(a) where a is n^2 - n - 1 ]; // _Vincenzo Librandi_, Aug 01 2011

%o (Haskell)

%o a002327 n = a002327_list !! (n-1)

%o a002327_list = filter ((== 1) . a010051') a028387_list

%o -- _Reinhard Zumkeller_, Jul 17 2014

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

%Y Cf. A010051.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E Extended by _Ray Chandler_, Sep 07 2005

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 13:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)