%I M4363 N1828 #203 Apr 06 2024 20:31:02

%S 7,19,37,61,127,271,331,397,547,631,919,1657,1801,1951,2269,2437,2791,

%T 3169,3571,4219,4447,5167,5419,6211,7057,7351,8269,9241,10267,11719,

%U 12097,13267,13669,16651,19441,19927,22447,23497,24571,25117,26227,27361,33391

%N Cuban primes: primes which are the difference of two consecutive cubes.

%C Primes of the form p = (x^3 - y^3)/(x - y) where x=y+1. See A007645 for generalization. I first saw the name "cuban prime" in Cunningham (1923). Values of x are in A002504 and y are in A111251. - _N. J. A. Sloane_, Jan 29 2013

%C Prime hex numbers (cf. A003215).

%C Equivalently, primes of the form p=1+3k(k+1) (and then k=floor(sqrt(p/3))). Also: primes p such that n^2(p+n) is a cube for some n>0. - _M. F. Hasler_, Nov 28 2007

%C Primes p such that 4p = 1+3s^2 for some integer s (A121259). - _Michael Somos_, Sep 15 2005

%C This sequence is believed to be infinite. - _N. J. A. Sloane_, May 07 2020

%D Allan Joseph Champneys Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.

%D Allan Joseph Champneys Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.

%D J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problem 241 pp. 39; 179, Ellipses Paris 2004.

%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 David A. Corneth, <a href="/A002407/b002407.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H A. J. C. Cunningham, <a href="/A002407/a002407.pdf">On quasi-Mersennian numbers</a>, Mess. Math., 41 (1912), 119-146. [Annotated scan of page 144 only]

%H A. J. C. Cunningham, <a href="/A001912/a001912.pdf">Binomial Factorisations</a>, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

%H R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>.

%H G. L. Honaker, Jr., <a href="https://t5k.org/curios/page.php?curio_id=22949">Prime curio for 127</a>.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=3F49x2R9Bno">Nearly cubic primes.</a>, YouTube video, 2023.

%H Project Euler, <a href="https://projecteuler.net/problem=131">Problem 131: Prime cube partnership</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubanPrime.html">Cuban Prime</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Cuban_prime">Cuban prime</a>.

%F a(n) = 6*A000217(A111251(n)) + 1. - _Christopher Hohl_, Jul 01 2019

%F From _Rémi Guillaume_, Nov 07 2023: (Start)

%F a(n) = A003215(A111251(n)).

%F a(n) = (3*(2*A002504(n) - 1)^2 + 1)/4.

%F a(n) = (3*A121259(n)^2 + 1)/4.

%F a(n) = prime(A145203(n)). (End)

%e a(1) = 7 = 1+3k(k+1) (with k=1) is the smallest prime of this form.

%e a(10^5) = 1792617147127 since this is the 100000th prime of this form.

%t lst={}; Do[If[PrimeQ[p=(n+1)^3-n^3], AppendTo[lst, p]], {n, 10^2}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Aug 21 2008 *)

%t Select[Table[3x^2+3x+1,{x,100}],PrimeQ] (* or *) Select[Last[#]- First[#]&/@ Partition[Range[100]^3,2,1],PrimeQ] (* _Harvey P. Dale_, Mar 10 2012 *)

%t Select[Differences[Range[100]^3],PrimeQ] (* _Harvey P. Dale_, Jan 19 2020 *)

%o (PARI) {a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( c<n, m++; if( isprime(m) && issquare((4*m-1)/3), c++)); m)} /* _Michael Somos_, Sep 15 2005 */

%o (PARI)

%o A002407(n,k=1)=until(isprime(3*k*k+++1) && !n--,);3*k*k--+1

%o list_A2407(Nmax)=for(k=1,sqrt(Nmax/3),isprime(t=3*k*(k+1)+1) && print1(t",")) \\ _M. F. Hasler_, Nov 28 2007

%o (Magma) [a: n in [0..100] | IsPrime(a) where a is (3*n^2+3*n+1)]; // _Vincenzo Librandi_, Jan 20 2020

%o (Python)

%o from sympy import isprime

%o def aupto(limit):

%o alst, k, d = [], 1, 7

%o while d <= limit:

%o if isprime(d): alst.append(d)

%o k += 1; d = 1+3*k*(k+1)

%o return alst

%o print(aupto(34000)) # _Michael S. Branicky_, Jul 19 2021

%Y Cf. A000217, A002504, A003215, A111251, A113478, A145203.

%Y Cf. A002648 (with x=y+2), A003627, A007645, A201477, A334520.

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, Aug 08 2000

%E Entry revised by _N. J. A. Sloane_, Jan 29 2013