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A002407
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Cuban primes: primes which are the difference of two consecutive cubes.
(Formerly M4363 N1828)
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25
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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
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1,1
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COMMENTS
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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
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REFERENCES
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Allan Joseph Champneys Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.
Allan Joseph Champneys Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
A. J. C. Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146. [Annotated scan of page 144 only]
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
G. L. Honaker, Jr., Prime curio for 127
Eric Weisstein's World of Mathematics, Cuban Prime
Wikipedia, Cuban prime
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EXAMPLE
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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.
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MATHEMATICA
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lst={}; Do[If[PrimeQ[p=(n+1)^3-n^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 *)
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PROG
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(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 */
(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from James A. Sellers, Aug 08 2000
Entry revised by N. J. A. Sloane, Jan 29 2013
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STATUS
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approved
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