A111251 -id:A111251 - 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

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

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

A350090 a(n) is the number of indices i in the range 0 <= i <= n-1 such that A003215(n) - A003215(i) is an oblong number (A002378), where A003215 are the hex numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 2, 3, 1, 1, 1, 3, 1, 1, 3, 3, 1, 3, 3, 3, 3, 5, 1, 1, 1, 5, 1, 1, 3, 1, 3, 1, 7, 1, 3, 3, 1, 1, 3, 7, 1, 1, 3, 3, 1, 3, 3, 1, 1, 3, 3, 1, 3, 7, 1, 3, 7, 1, 7, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 7, 5, 3, 3, 1, 5, 3, 3, 7, 3, 1, 1, 3, 3, 3, 7, 1, 3, 1, 3, 1

Offset: 0

Klaus Purath and Michel Marcus, Dec 14 2021

nonn

There are very few even terms in the data (3 up to 10000). They are obtained for indices coming from A001921. For odd terms see A350120.

a(n) = 1 for n in A111251.

			For n=5, the 5 numbers hex(5)-hex(i), for i=0 to 4, are (90, 84, 72, 54, 30) out of which 90, 72 and 30 are oblong, so a(5) = 3.

Michel Marcus, Table of n, a(n) for n = 0..10000

Cf. A002378, A003215.

Cf. also A001921, A111251, A350120.

obQ[n_] := IntegerQ @ Sqrt[4*n + 1]; hex[n_] := 3*n*(n + 1) + 1; a[n_] := Module[{h = hex[n]}, Count[Range[0, n - 1], ?(obQ[h - hex[#]] &)]]; Array[a, 100, 0] (* _Amiram Eldar, Dec 14 2021 *)

hex(n) = 3*n*(n+1)+1; \\ A003215
isob(n) = my(m=sqrtint(n)); m*(m+1)==n; \\ A002378
a(n) = my(h=hex(n)); sum(k=0, n-1, isob(h - hex(k)));

a(n) = numdiv(3*n*n + 3*n + 1) - 1; \\ Jinyuan Wang, Dec 19 2021

a(n) = A000005(A003215(n)) - 1. - Jinyuan Wang, Dec 19 2021

Edited by N. J. A. Sloane, Dec 25 2021

A376907 a(n) is the least n-digit cuban prime.

Original entry on oeis.org

7, 19, 127, 1657, 10267, 102121, 1021417, 10052191, 100381321, 1000556719, 10000510297, 100025541019, 1000011191887, 10000028937841, 100000062634561, 1000001305386991, 10000001240507791, 100000021541868691, 1000000084213608427, 10000000012591553221, 100000000159478313337

Offset: 1

Stefano Spezia, Oct 08 2024

a(n) - A011557(n-1) is a multiple of 3.

Robert Israel, Table of n, a(n) for n = 1..996

Cf. A002407, A003215, A003617, A011557, A111251, A113478, A145203, A376933.

nextcuban:= proc(n)
  local k,y;
  for k from ceil((sqrt(12*n-3)-3)/6) do
    y:= (k+1)^3 - k^3;
    if isprime(y) then return y fi
  od
end proc:
seq(nextcuban(10^i), i = 0 .. 25); # Robert Israel, Nov 08 2024

a[n_]:=Module[{k=1},While[!PrimeQ[m=3k^2+3k+1]||IntegerLength[m]

from itertools import count
from math import isqrt
from sympy import isprime
def A376907(n):
    for k in count(isqrt((((a:=10**(n-1))<<2)-1)//12)):
        m = 3*k*(k+1)+1
        if m >= a and isprime(m):
            return m # Chai Wah Wu, Oct 13 2024

Conjecture: a(n+1)/a(n) ~ 10.

A257772 Numbers n>=0 such that (n+1)^3 - n^3 = 3n^2+3n+1 is not prime.

Original entry on oeis.org

0, 5, 7, 8, 12, 15, 16, 18, 19, 20, 21, 22, 26, 29, 31, 33, 35, 36, 39, 40, 43, 44, 46, 47, 50, 51, 53, 54, 56, 57, 59, 60, 61, 64, 65, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 82, 83, 84, 85, 87, 89, 92, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 106

Offset: 1

Miguel Angel Velilla Mula, May 08 2015

Complement of A111251.

Includes all members of A047383 except 1. - Robert Israel, May 12 2015

			5 is a term since (5+1)^3 - 5^3 = 91 = 13*7 is not prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A003215, A047383, A111251.

[n: n in [0..120] | not IsPrime(3*n^2+3*n+1)]; // Vincenzo Librandi, May 13 2015

remove(t -> isprime((t+1)^3-t^3), [$0..300]); # Robert Israel, May 12 2015

Select[Range[0, 200], ! PrimeQ[(#+1)^3 - #^3] &] (* Giovanni Resta, May 08 2015 *)

for(n=0,100,if(!isprime(3*n^2+3*n+1),print1(n,", "))) \\ Derek Orr, May 19 2015

10 print 0
20 for n=1 to 200
30   s = (n+1)^3 - n^3
40   if prmdiv(s)<>s then print n
50 next n

a(n) ~ n. - Charles R Greathouse IV, May 22 2015

A338610 Integers m such that there exist one prime p and one positive integer k, for which the expression k^3 + k^2*p is a perfect cube m^3.

Original entry on oeis.org

2, 12, 36, 80, 252, 810, 1100, 1452, 2366, 2940, 5202, 12696, 14400, 16250, 20412, 22736, 27900, 33792, 40460, 52022, 56316, 70602, 75852, 93150, 112896, 120050, 143312, 169400, 198476, 242172, 254016, 291852, 305252, 410700, 518400, 538002, 643452, 689216, 737100

Offset: 1

Bernard Schott, Nov 03 2020

nonn

This concerns Problem 131 of Project Euler (see link).
For each such term m with this property, the values of k and of p are unique.
The solution to the Diophantine equation is: (q^3)^3 + (q^3)^2 * ((q+1)^3 - q^3) = ((q+1) * q^2)^3, where
- the prime p is the cuban prime (q+1)^3 - q^3 = A002407(n),
- corresponding to q = A111251(n),
- the positive integer k = q^3, and,
- the resulting m = (q+1)*q^2 = (A111251(n)+1)*(A111251(n))^2.

			For n=1, q=A111251(1)=1 and 1^3 + 1^2*(2^3 - 1^3) = 1+1*7 = 2^3, hence, k=1^3, cuban prime=7, and a(1)=m=2.
For n=3, q=A111251(3)=3 and (3^3)^3 + (3^3)^2*(4^3 - 3^3) = 27^3 + 27^2*37 = 46656 = 36^3, hence, k=3^3, cuban prime=37, and a(3)=m=36.

Robert Israel, Table of n, a(n) for n = 1..10000
Project Euler, Problem 131: Prime cube partnership.

Cf. A002407, A111251.

Subsequence of A011379.

for q from 1 to 90 do
p:=3*q^2+3*q+1;
if isprime(p) then print((q+1)*q^2); else fi; od:

f[n_] := n^2*(n+1); f /@ Select[Range[100], PrimeQ[3*#^2 + 3*# + 1] &] (* Amiram Eldar, Nov 05 2020 *)

lista(nn) =  apply(x->x^2*(x+1), select(x->isprime(3*x^2 + 3*x + 1), [1..nn])); \\ Michel Marcus, Nov 05 2020

a(n) = (A111251(n) + 1)*(A111251(n))^2.

a(n) = A011379(A111251(n)).

A111292 Numbers n such that 6n^2 + 6n + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 18, 19, 20, 22, 23, 30, 31, 32, 33, 35, 36, 38, 41, 42, 43, 45, 46, 47, 51, 55, 58, 60, 65, 67, 73, 74, 77, 78, 84, 86, 88, 93, 95, 97, 100, 101, 104, 106, 107, 109, 112, 117, 120, 123, 124, 126, 129, 130, 132, 134, 135, 137, 143, 148, 151

Offset: 1

Parthasarathy Nambi, Nov 01 2005

			If n=43 then 6*n^2 + 6*n + 1 = 11353 (prime).

Cf. A111251, A090563.

[n: n in [0..200] |IsPrime(6*n^2+6*n+1)]; // Vincenzo Librandi, Nov 13 2010

is(n)=isprime(6*n^2+6*n+1) \\ Charles R Greathouse IV, Jun 13 2017

cp's OEIS Frontend

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

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

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

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A350090 a(n) is the number of indices i in the range 0 <= i <= n-1 such that A003215(n) - A003215(i) is an oblong number (A002378), where A003215 are the hex numbers.

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A376907 a(n) is the least n-digit cuban prime.

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A257772 Numbers n>=0 such that (n+1)^3 - n^3 = 3*n^2+3*n+1 is not prime.

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

A257772 Numbers n>=0 such that (n+1)^3 - n^3 = 3n^2+3n+1 is not prime.

A111292 Numbers n such that 6n^2 + 6n + 1 is prime.