A121259 -id:A121259 - 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

A002504 Numbers x such that 1 + 3x(x-1) is a ("cuban") prime (cf. A002407).

Original entry on oeis.org

2, 3, 4, 5, 7, 10, 11, 12, 14, 15, 18, 24, 25, 26, 28, 29, 31, 33, 35, 38, 39, 42, 43, 46, 49, 50, 53, 56, 59, 63, 64, 67, 68, 75, 81, 82, 87, 89, 91, 92, 94, 96, 106, 109, 120, 124, 126, 129, 130, 137, 141, 143, 148, 154, 157, 158, 159, 165, 166, 171, 172

Offset: 1

N. J. A. Sloane

nonn

Equivalently, positive integers x such that x^3 - (x-1)^3 is prime. - Rémi Guillaume, Oct 24 2023

			From _Rémi Guillaume_, Dec 07 2023: (Start)
1 + 3*7*6 = 127 = A002407(5) is the 5th prime of this form, so a(5) = 7.
1 + 3*10*9 = 271 = A002407(6) is the 6th prime of this form, so a(6) = 10.
(End)

A. J. C. Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002407 (resulting primes), A111251, A121259.

Select[Range[500], PrimeQ[1 + 3 # (# - 1)] &] (* T. D. Noe, Jan 30 2013 *)

for(k=1,999,isprime(3*k*(k-1)+1)&print1(k",")) \\ M. F. Hasler, Nov 28 2007

From Rémi Guillaume, Dec 07 2023: (Start)
a(n) = ceiling(sqrt(A002407(n)/3)).
a(n) = A111251(n) + 1.
a(n) = (A121259(n) + 1)/2. (End)

Edited, updated (1 is no longer regarded as a prime) and extended by M. F. Hasler, Nov 28 2007

A111051 Numbers m such that 3*m^2 + 1 is prime.

Original entry on oeis.org

2, 6, 8, 12, 16, 20, 22, 26, 34, 36, 40, 58, 64, 68, 78, 82, 84, 86, 98, 112, 120, 126, 142, 146, 148, 152, 156, 160, 168, 188, 190, 194, 196, 208, 216, 218, 222, 238, 240, 244, 246, 254, 264, 272, 282, 286, 294, 300, 302, 306, 308, 316, 320, 330, 338, 344, 348

Offset: 1

Parthasarathy Nambi, Oct 06 2005

nonn

The resulting primes are the generalized cuban primes of the form (x^3-y^3)/(x-y), x=y+2 (see A002648). - Jani Melik, Jul 18 2007

			1 + 3*2^2 = 13 = A002648(1) is the 1st prime of this form, so a(1) = 2.
1 + 3*6^2 = 109 = A002648(2) is the 2nd prime of this form, so a(2) = 6.
1 + 3*8^2 = 193 = A002648(3) is the 3rd prime of this form, so a(3) = 8.
If m=98 then 3*m^2 + 1 = 28813 = A002648(19) is prime (the 19th of this form), so 98 is a term (the 19th).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A002648, A121259.

ts_kubpra_ind:=proc(n) local i, tren, ans; ans:=[ ]: for i from 0 to n do tren:=1+3*i^2: if (isprime(tren)='true') then ans:=[ op(ans),i ] fi od: RETURN(ans); end: ts_kubpra_ind(2000); # Jani Melik, Jul 18 2007

Select[Range[400],PrimeQ[3#^2+1]&] (* Harvey P. Dale, Jul 17 2016 *)

is(n)=isprime(3*n^2+1) \\ Charles R Greathouse IV, Feb 07 2017

a(n) = sqrt((A002648(n)-1)/3). - Zak Seidov, Feb 04 2016

More terms from Jani Melik, Jul 18 2007

Edited by N. J. A. Sloane, Sep 28 2007

A120460 Primes p such that (3*p^2+1)/4 is prime.

Original entry on oeis.org

3, 5, 7, 13, 19, 23, 29, 47, 61, 83, 97, 127, 149, 163, 173, 181, 191, 211, 239, 251, 257, 281, 307, 313, 317, 331, 359, 373, 383, 419, 433, 439, 449, 467, 491, 503, 593, 607, 617, 631, 643, 701, 709, 719, 751, 797, 811, 839, 859, 883, 887, 937, 971, 1013, 1049

Offset: 1

Zak Seidov, Aug 25 2006

Prime terms in A121259 = numbers n such that (3n^2+1)/4 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A121259.

Select[Prime[Range[200]],PrimeQ[(3#^2+1)/4]&] (* Harvey P. Dale, Apr 25 2017 *)

A377045 Number of partitions of cuban primes.

Original entry on oeis.org

15, 490, 21637, 1121505, 3913864295, 1131238503938606, 78801255302666615, 5589233202595404488, 29349508915133986374841, 2163909235608484556362424, 913865816485680423486405066750, 191623400974625892978847721669762887224010

Offset: 1

Paul F. Marrero Romero, Oct 14 2024

nonn

Number of partitions of prime numbers that are the difference of two consecutive cubes.

Number of partitions of primes p such that p=(3*k^2 + 1)/4 for some integer k (A121259).

Robert Israel, Table of n, a(n) for n = 1..153

Cf. A000041, A002407, A121259.

R:= NULL: count:= 0:
for i from 1 while count < 30 do
  p:= (i+1)^3 - i^3;
  if isprime(p) then count:= count+1; v:= combinat:-numbpart(p); R:= R,v; fi
od:
R; # Robert Israel, Nov 14 2024

PartitionsP[Select[Table[(3k^2 + 1)/4,{k,50}],PrimeQ]]

a(n) = A000041(A002407(n)).

a(n) = A000041((3*A121259(n)^2 + 1)/4).

cp's OEIS Frontend

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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A002504 Numbers x such that 1 + 3*x*(x-1) is a ("cuban") prime (cf. A002407).

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A111051 Numbers m such that 3*m^2 + 1 is prime.

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A120460 Primes p such that (3*p^2+1)/4 is prime.

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A377045 Number of partitions of cuban primes.

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Author

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Programs

Maple

Mathematica

Formula

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

A002504 Numbers x such that 1 + 3x(x-1) is a ("cuban") prime (cf. A002407).