A002407 Cuban primes: primes which are the difference of two consecutive cubes. (Formerly M4363 N1828)

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, 27361, 33391

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

This sequence is believed to be infinite. - N. J. A. Sloane, May 07 2020

References

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 Problèmes en Théorie 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

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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]

R. K. Guy, Letter to N. J. A. Sloane, 1987

G. L. Honaker, Jr., Prime curio for 127

Michael Penn, Nearly cubic primes., YouTube video, 2023.

Project Euler, Problem 131: Prime cube partnership.

Eric Weisstein's World of Mathematics, Cuban Prime

Wikipedia, Cuban prime

Formula

a(n) = 6*A000217(A111251(n)) + 1. - Christopher Hohl, Jul 01 2019

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)^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 *)
Select[Differences[Range[100]^3], PrimeQ] (* Harvey P. Dale, Jan 19 2020 *)

Prog

(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
(Magma) [a: n in [0..100] | IsPrime(a) where a is (3*n^2+3*n+1)]; // Vincenzo Librandi, Jan 20 2020
(Python)
from sympy import isprime
def aupto(limit):
    alst, k, d = [], 1, 7
    while d <= limit:
        if isprime(d): alst.append(d)
        k += 1; d = 1+3*k*(k+1)
    return alst
print(aupto(34000)) # Michael S. Branicky, Jul 19 2021

Crossrefs

Cf. A002504, A003215, A002648, A007645, A003627, A113478, A201477.

Cf. A000217, A111251.

See also A334520.

Sequence in context: A003215 A308685 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