A002407 Cuban primes: primes which are the difference of two consecutive cubes.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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 and y are in A111251. - 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+3s^2 for some integer s (A121259). - Michael Somos, Sep 15 2005

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

Examples

			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.
		

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.

Crossrefs

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

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

Programs

Magma

[a: n in [0..100] | IsPrime(a) where a is (3*n^2+3*n+1)]; // Vincenzo Librandi, Jan 20 2020

Mathematica

lst={}; Do[If[PrimeQ[p=(n+1)^3-n^3], 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 *)

PARI

{a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( cMichael 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

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

Formula

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

From Rémi Guillaume, Nov 07 2023: (Start)

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

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

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

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

Extensions

More terms from James Sellers, Aug 08 2000

Entry revised by N. J. A. Sloane, Jan 29 2013