A002504 -id:A002504 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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.

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).

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.

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

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

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

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 *)

{a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( cMichael Somos, Sep 15 2005 */

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

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

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)

More terms from James Sellers, Aug 08 2000

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

A111251 Numbers k such that 3k^2 + 3k + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 10, 11, 13, 14, 17, 23, 24, 25, 27, 28, 30, 32, 34, 37, 38, 41, 42, 45, 48, 49, 52, 55, 58, 62, 63, 66, 67, 74, 80, 81, 86, 88, 90, 91, 93, 95, 105, 108, 119, 123, 125, 128, 129, 136, 140, 142, 147, 153, 156, 157, 158, 164, 165, 170, 171, 172, 175

Offset: 1

Parthasarathy Nambi, Oct 31 2005

nonn

That is, positive integers k such that (k+1)^3 - k^3 is prime.
The Hardy-Littlewood constant 1.68109913... of this polynomial is approximately half that of the well-known Euler polynomial A221712, i.e., in comparison, only about half as many prime numbers are produced asymptotically as with k^2 + k + 41. - Hugo Pfoertner, Feb 10 2020
The primes that are obtained are called cuban primes and are in A002407. - Bernard Schott, Feb 13 2020

			For k=52, 3*52^2 + 3*52 + 1 = 8269 is prime, so 52 is a term.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000 (terms 1..1965 from Pierre CAMI)

Cf. A221712, A002407 (resulting primes), A002504, A121259.

[k: k in [1..180] | IsPrime(3*k^2 + 3*k + 1)]; // Marius A. Burtea, Feb 10 2020

Select[Range[200],PrimeQ[3#^2+3#+1]&] (* Harvey P. Dale, May 29 2017 *)

for(n=0,250,if(isprime(3*n^2+3*n+1),print1(n,",")))

a(n) = floor(sqrt(A002407(n)/3)). - Rémi Guillaume, Oct 16 2023
a(n) = A002504(n) - 1. - Rémi Guillaume, Oct 21 2023
a(n) = (A121259(n) - 1)/2. - Rémi Guillaume, Dec 29 2023

Extended by Lambert Klasen (lambert.klasen(AT)gmx.net), Nov 02 2005

A113478 Number of cuban primes less than 10^n.

Original entry on oeis.org

0, 1, 4, 11, 28, 64, 173, 438, 1200, 3325, 9289, 26494, 76483, 221530, 645685, 1895983, 5593440, 16578830, 49347768, 147402214, 441641536, 1326941536, 3996900895, 12066234206, 36501753353

Offset: 0

Eric W. Weisstein, Jan 10 2006

			7, 19, 37, 61, 127 are the first few cuban primes, so a(1)=1 and a(2)=4.

Eric Weisstein's World of Mathematics, Cuban Prime

Cf. A002407, A002504.

A113478(n)=sum(k=1,sqrt(10^n/3),isprime(3*k*(k+1)+1)) \\ M. F. Hasler, Nov 28 2007

a(15)-a(18) from Donovan Johnson, Feb 05 2010

a(19)-a(24) from Hiroaki Yamanouchi, Oct 08 2015

A121259 Numbers k such that (3*k^2 + 1)/4 is prime.

Original entry on oeis.org

3, 5, 7, 9, 13, 19, 21, 23, 27, 29, 35, 47, 49, 51, 55, 57, 61, 65, 69, 75, 77, 83, 85, 91, 97, 99, 105, 111, 117, 125, 127, 133, 135, 149, 161, 163, 173, 177, 181, 183, 187, 191, 211, 217, 239, 247, 251, 257, 259, 273, 281, 285, 295, 307, 313, 315, 317, 329, 331, 341

Offset: 1

Zak Seidov, Aug 23 2006

			(3*5^2 + 1)/4 = 19 is the 2nd prime of this form, so a(2) = 5.
(3*13^2 + 1)/4 = 127 is the 5th prime of this form, so a(5) = 13.
(3*19^2 + 1)/4 = 271 is the 6th prime of this form, so a(6) = 19.

Michel Marcus, Table of n, a(n) for n = 1..8144

Cf. comment by Michael Somos in A002407.

Cf. A002504, A111251, A111051 (simpler variant).

Select[Range[400],PrimeQ[(3#^2+1)/4]&]  (* Harvey P. Dale, Mar 24 2011 *)

is(n) = my(p=(3*n^2+1)/4); (denominator(p)==1) && isprime(p); \\ Charles R Greathouse IV, Jun 13 2017; edited by Michel Marcus, Oct 12 2023

a(n) = sqrt((4*A002407(n) - 1)/3). [corrected by Rémi Guillaume, Dec 07 2023]
a(n) = 2*A002504(n) - 1. - Hugo Pfoertner, Oct 07 2023
a(n) = 2*A111251(n) + 1. - Rémi Guillaume, Dec 06 2023

A221794 Number of primes of the form (x+1)^3 - x^3 with x <= 10^n.

Original entry on oeis.org

1, 7, 42, 263, 1965, 15282, 126826

Offset: 0

Vladimir Pletser, Jan 25 2013

nonn

Cuban primes are primes that are the difference of two consecutive cubes, p = (x+1)^3 - x^3 (A002407). They are also primes of the form 3k(k+1) + 1, where values for k+1 are in A002504.

Ed Pegg, Jr., MathWorld: Cuban Prime
Wikipedia, Cuban prime

Cf. A002407, A002504, A003215, A113478 (number of cuban primes < 10^n).

A066486 a(n) = min( x : x^3 + n^3 == 0 mod (x+n-1) ).

Original entry on oeis.org

1, 6, 17, 34, 57, 2, 121, 6, 23, 262, 321, 386, 55, 534, 617, 88, 3, 902, 61, 144, 77, 52, 9, 1634, 1777, 1926, 17, 2242, 2409, 344, 2761, 198, 3137, 4, 3537, 164, 535, 4182, 4409, 112, 93, 5126, 5377, 768, 413, 6166, 453, 920, 7009, 7302, 1043, 22, 8217, 224, 13, 9186, 5, 34, 10209, 188, 19, 1560, 11657

Offset: 1

Benoit Cloitre, Jan 02 2002

nonn

Cf. A066333.

a[n_] := For[x = 1, True, x++, If[Mod[x^3 + n^3, x + n - 1] == 0, Return[x]]]; Array[a, 24] (* Jean-François Alcover, Feb 17 2018 *)

a(n) = {my(k=1); while((k^3+n^3)%(k+n-1) != 0, k++); k; } \\ Altug Alkan, Feb 17 2018

a(n) = 3*n^2 - 4*n + 2 for n=1, 2, 3, 4, 5, 7, 10, 11, 12, 14, 15, 18, 24, 25, 26, 28, 29, 31, 33, 35, 38, ...

That is, in those cases a(n) = A056109(n-1). It appears that the corresponding indices are given by A133431 (i.e., 1 U A002504). - Michel Marcus, Feb 17 2018

More terms from Altug Alkan, Feb 17 2018

cp's OEIS Frontend

A133431 Old-fashioned version of A002504 (the initial 1 should be omitted since 1 is no longer regarded as a prime, although it was in 1912).

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

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A111251 Numbers k such that 3*k^2 + 3*k + 1 is prime.

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A113478 Number of cuban primes less than 10^n.

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A121259 Numbers k such that (3*k^2 + 1)/4 is prime.

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A221794 Number of primes of the form (x+1)^3 - x^3 with x <= 10^n.

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A066486 a(n) = min( x : x^3 + n^3 == 0 mod (x+n-1) ).

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A111251 Numbers k such that 3k^2 + 3k + 1 is prime.