|
|
A002407
|
|
Cuban primes: primes which are the difference of two consecutive cubes.
(Formerly M4363 N1828)
|
|
31
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
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
|
|
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
|
A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
|
|
FORMULA
|
|
|
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
|
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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|