A113478 -id:A113478 - 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

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).

A221792 Number of n-digit cuban primes.

Original entry on oeis.org

1, 3, 7, 17, 36, 109, 265, 762, 2125, 5964, 17205, 49989, 145047, 424155, 1250298, 3697457, 10985390, 32768938, 98054446, 294239322, 885300000, 2669959359, 8069333311, 24435519147

Offset: 1

Vladimir Pletser, Jan 25 2013

Cf. A002407, A113478, A221793.

a(n) = A113478(n) - A113478(n-1). - Jens Kruse Andersen, Jul 14 2014

a(19)-a(24) added from A113478 by Andrew Howroyd, Jan 14 2020

A221846 Number of primes of the form (x+1)^5 - x^5 less than 10^n.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 12, 19, 30, 40, 66, 110, 173, 285, 463, 749, 1256, 2075, 3499, 5884, 9928, 16754, 28345, 48037, 82187, 140358, 239768, 409315, 700510, 1200863, 2061093, 3544072, 6098353, 10505051

Offset: 0

Vladimir Pletser, Jan 26 2013

nonn

Number of primes less than 10^n and equal to the difference of two consecutive fifth powers (x+1)^5 - x^5 = 5x(x+1)(x^2+x+1)+1 (A121616). Values of x = A121617. Sequence of number of primes less than 10^n and of the form (x+1)^5 - x^5 have similar characteristics to similar sequences for natural primes (A006880) and cuban primes (A113478).

A221977 Number of primes of the form (x+1)^7 - x^7 less than 10^n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 7, 8, 10, 14, 18, 25, 34, 46, 60, 89, 120, 165, 227, 298, 415, 590, 821, 1152, 1606, 2240, 3188, 4438, 6208, 8714, 12280, 17368, 24560, 34821, 49413, 70581, 100856, 143955, 205291, 293061, 419256, 600213, 858870, 1230523, 1764914, 2532078

Offset: 1

Vladimir Pletser, Feb 02 2013

Number of primes less than 10^n and equal to the difference of two consecutive seventh powers (x+1)^7 - x^7 = 7x(x+1)(x^2+x+1)^2+1 (A121618). Values of x = A121619 - 1. Sequence of number of primes less than 10^n and of the form (x+1)^7 - x^7 have similar characteristics to similar sequences for natural primes (A006880), cuban primes (A113478) and primes of the form (x+1)^5 - x^5 (A221846).

Vladimir Pletser, Table of n, a(n) for n = 1..50

nn = 20; t = Table[0, {nn}]; n = 0; While[n++; p = (n + 1)^7 - n^7; p < 10^nn,If[PrimeQ[p], m = Ceiling[Log[10, p]]; t[[m]]++]]; Accumulate[t] (* T. D. Noe, Feb 04 2013 *)

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.

A221983 Number of primes of the form (x+1)^11 - x^11 less than 10^n.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 5, 6, 6, 8, 9, 9, 9, 12, 15, 18, 24, 30, 35, 41, 46, 66, 83, 104, 133, 166, 195, 247, 314, 400, 475, 589, 709, 855, 1046, 1313, 1604, 1998, 2468, 3029, 3681, 4518, 5581, 6920, 8629, 10647, 13122, 16214, 19894, 24644, 30569, 37864, 46927

Offset: 9

Vladimir Pletser, Feb 02 2013

nonn

Number of primes less than 10^n and equal to the difference of two consecutive eleventh powers (x+1)^11 - x^11 = 11x(x+1)(x^2+x+1)( x(x+1)(x^2+x+1)(x^2+x+3)+1) + 1 (A189055). Values of x = A211184. Sequence of number of primes less than 10^n and of the form (x+1)^11 - x^11 have similar characteristics to similar sequences for natural primes (A006880), cuban primes (A113478) and primes of the form (x+1)^p - x^p for p = 5 (A221846) and p = 7 (A221977).

Vladimir Pletser, Table of n, a(n) for n = 9..86

nn = 40; t = Table[0, {nn}]; n = 0; While[n++; p = (n + 1)^11 - n^11; p < 10^nn, If[PrimeQ[p], m = Ceiling[Log[10, p]]; t[[m]]++]]; Accumulate[t] (* T. D. Noe, Feb 04 2013 *)

A210520 Number of cuban primes < 10^(n/2).

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 11, 17, 28, 42, 64, 105, 173, 267, 438, 726, 1200, 2015, 3325, 5524, 9289, 15659, 26494, 44946, 76483, 129930, 221530, 377856, 645685, 1105802, 1895983, 3254036, 5593440, 9625882, 16578830, 28590987, 49347768, 85253634

Offset: 0

Vladimir Pletser, Jan 26 2013

nonn

A cuban prime has the form (x+1)^3 - x^3, which equals 3x*(x+1) + 1 (A002407).

			As the smallest cuban primes equal to the difference of two consecutive cubes p = (x+1)^3 - x^3, is 7 for x = 1, and as floor (10^(1/2)) = 3, a(0) = a(1) = 0 and a(2) = 1.

Cf. A002407, A113478, A221794.

cnt = 0; nxt = 1; t = {0}; Do[p = 3*k*(k + 1) + 1; If[p > nxt, AppendTo[t, cnt]; nxt = nxt*Sqrt[10]]; If[PrimeQ[p], cnt++], {k, 100000}]; t (* T. D. Noe, Jan 29 2013 *)

b(n)={my(s=0,k=0,t=1); while(t<=n, s+=isprime(t); k++; t += 6*k); s}
a(n)={b(sqrtint(10^n))} \\ Andrew Howroyd, Jan 14 2020

a(2*n) = A113478(n). - Andrew Howroyd, Jan 14 2020

a(31)-a(37) from Andrew Howroyd, Jan 14 2020

cp's OEIS Frontend

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

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

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A221792 Number of n-digit cuban primes.

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A221846 Number of primes of the form (x+1)^5 - x^5 less than 10^n.

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A221977 Number of primes of the form (x+1)^7 - x^7 less than 10^n.

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

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A221983 Number of primes of the form (x+1)^11 - x^11 less than 10^n.

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A210520 Number of cuban primes < 10^(n/2).

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