

A002407


Cuban primes: primes which are the difference of two consecutive cubes.
(Formerly M4363 N1828)


22



7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227
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OFFSET

1,1


COMMENTS

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.  N. J. A. Sloane, Jan 29 2013
Prime hex numbers (cf. A003215).
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
Primes p such that 4p = 1+3n^2 for some integer n.  Michael Somos, Sep 15 2005
The cuban primes may be generated from the hexagonal centered numbers by eliminating all the items that may be expressed as 36*i*j + 6*i + 6*j + 1 with i,j integers.  Giacomo Fecondo, Mar 13 2009, Mar 17 2009


REFERENCES

A. J. C. Cunningham, On quasiMersennian numbers, Mess. Math., 41 (1912), 119146.
A. J. C. Cunningham, Binomial Factorisations, Vols. 19, Hodgson, London, 19231929; see Vol. 1, pp. 245259.
J.M. De Koninck & A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 241 pp. 39; 179, Ellipses Paris 2004.
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

T. D. Noe, Table of n, a(n) for n=1..1000
G. L. Honaker, Jr., Prime curio for 127
Eric Weisstein's World of Mathematics, Cuban Prime
Wikipedia, Cuban prime


EXAMPLE

a(1) = 7 = 1+3k(k+1) (with k=1) is the smallest prime of this form.
a(10^5) = 1792617147127 since this is the 100000th prime of this form.


MATHEMATICA

lst={}; Do[If[PrimeQ[p=(n+1)^3n^3], (*Print[p]; *)AppendTo[lst, p]], {n, 10^2}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
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 *)


PROG

(PARI) {a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( c<n, m++; if( isprime(m)&issquare((4*m1)/3), c++)); m)} /* Michael Somos, Sep 15 2005 */
(PARI) A002407(n, k=1)=until(isprime(3*k*k+++1)&!n, ); 3*k*k+1 list_A2407(Nmax)=for(k=1, sqrt(Nmax/3), isprime(t=3*k*(k+1)+1)&print1(t", ")) \\ M. F. Hasler, Nov 28 2007


CROSSREFS

Cf. A002504, A003215, A002648, A007645, A003627, A113478, A201477.
Sequence in context: A113743 A003215 A133323 * A098484 A155443 A155405
Adjacent sequences: A002404 A002405 A002406 * A002408 A002409 A002410


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from James A. Sellers, Aug 08 2000
Entry revised by N. J. A. Sloane, Jan 29 2013


STATUS

approved



