A236475 -id:A236475 - cp's OEIS frontend

A236526 Numbers k such that k^3 + k +- 1 are twin primes.

3, 15, 18, 21, 39, 87, 117, 120, 135, 243, 360, 366, 381, 426, 429, 615, 642, 723, 879, 1002, 1023, 1170, 1173, 1224, 1458, 1506, 1518, 1530, 1731, 1896, 1920, 1965, 2007, 2025, 2058, 2133, 2160, 2376, 2379, 2382, 2406, 2553, 2577, 2673, 2703, 2727

Offset: 1

Derek Orr, Jan 27 2014

nonn

The only prime in this sequence is a(1) = 3.

			381^3 + 381 +- 1 (55305961 and 55305959, respectively) are both prime. Thus, 381 is a member of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A049407 and A236475.

[n: n in [1..5*10^3] |IsPrime(n^3+n-1) and IsPrime(n^3 +n+1)]; // Vincenzo Librandi, Dec 26 2015

Select[Range[3000], PrimeQ[#^3 + # - 1] && PrimeQ[#^3 + # + 1] &] (* Vincenzo Librandi, Dec 26 2015 *)
Select[Range[3000],AllTrue[#^3+#+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Feb 23 2020 *)

isok(n) = isprime(n^3+n+1) && isprime(n^3+n-1); \\ Michel Marcus, Dec 27 2015

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n**3+n-1) and isprime(n**3+n+1)}

A236764 Numbers k such that k^3 +/- k +/- 1 are prime for all four possibilities.

Original entry on oeis.org

15, 21, 15375, 25164, 53361, 95190, 110685, 115140, 133701, 139425, 140430, 140844, 189336, 217686, 220650, 266916, 272469, 289341, 344880, 364665, 377805, 382221, 390270, 415779, 454905, 539700, 561186, 567645, 575799, 584430, 603651, 722484

Offset: 1

Derek Orr, Jan 30 2014

nonn

			110685^3+110685+1 (1356020665779811), 110685^3+110685-1 (1356020665779809), 110685^3-110685+1 (1356020665558441) and 110685^3-110685-1 (1356020665558439) are all prime. Thus 110685 is a member of this sequence.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Intersection of A126421, A236477, A049407, and A236475.

for(n=1, 800000, if(isprime(n^3+n+1)&&isprime(n^3-n+1)&&isprime(n^3+n-1)&&isprime(n^3-n-1), print1(n, ","))) \\ Colin Barker, Jan 31 2014

import sympy
from sympy import isprime
{print(n) for n in range(10**6) if isprime(n**3+n+1) and isprime(n**3-n+1) and isprime(n**3+n-1) and isprime(n**3-n-1)}

A236476 Primes p such that p^3 + p - 1 is prime.

Original entry on oeis.org

3, 7, 43, 73, 103, 109, 127, 139, 151, 199, 223, 241, 283, 313, 367, 379, 421, 541, 631, 661, 733, 739, 751, 769, 829, 991, 1117, 1129, 1201, 1231, 1249, 1297, 1303, 1429, 1471, 1663, 1669, 1693, 1699, 1741, 1789, 1867, 1933

Offset: 1

Derek Orr, Jan 26 2014

nonn

Primes in the sequence A236475.

			241 is prime and 241^3 + 241 - 1 = 13997761 is prime.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A236475.

s=[]; forprime(p=2, 2000, if(isprime(p^3+p-1), s=concat(s, p))); s \\ Colin Barker, Jan 27 2014

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n) and isprime(n**3+n-1)}

A248079 Least number k such that k^n + k - 1 is prime, or 0 if no such k exists.

Original entry on oeis.org

2, 2, 3, 2, 0, 4, 6, 2, 4, 3, 0, 17, 36, 3, 3, 2, 0, 6, 9, 43, 27, 9, 0, 3, 154, 3, 34, 54, 0, 24, 24, 6, 226, 23, 0, 3, 70, 36, 13, 51, 0, 4, 13, 9, 10, 68, 0, 18, 10, 45, 154, 85, 0, 23, 6, 10, 37, 8, 0, 30, 331, 9, 3, 40, 0, 11, 61, 8, 10, 35, 0, 61, 76, 54, 426, 9, 0, 84, 87, 13, 46

Offset: 1

Derek Orr, Sep 30 2014

nonn

If n == 5 mod 6 (A016969), k^n + k - 1 is always divisible by k^2 - k + 1. Thus it will never be prime.

Cf. A016969, A045546, A236475, A236759, A113517.

lnk[n_]:=Module[{k=2},While[CompositeQ[k^n+k-1],k++];k]; Table[If[Mod[n,6] == 5,0,lnk[n]],{n,90}] (* Harvey P. Dale, Oct 24 2021 *)

a(n)=if(n==Mod(5,6),return(0));k=1;while(!isprime(k^n+k-1),k++);k
vector(100,n,a(n))

cp's OEIS Frontend

A236526 Numbers k such that k^3 + k +- 1 are twin primes.

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A236764 Numbers k such that k^3 +/- k +/- 1 are prime for all four possibilities.

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A236476 Primes p such that p^3 + p - 1 is prime.

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Author

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Examples

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Crossrefs

Programs

PARI

Python

A248079 Least number k such that k^n + k - 1 is prime, or 0 if no such k exists.

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Author

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Programs

Mathematica

PARI