A236759 -id:A236759 - cp's OEIS frontend

A236760 Numbers n such that n^4 + n +- 1 are twin primes.

2, 6, 9, 12, 26, 44, 72, 77, 119, 204, 266, 290, 351, 506, 539, 542, 561, 644, 741, 807, 861, 924, 992, 996, 1016, 1032, 1049, 1356, 1412, 1556, 1640, 1794, 1847, 1862, 1871, 1895, 1980, 2036, 2129, 2222, 2289, 2354, 2445, 2616, 2630

Offset: 1

Derek Orr, Jan 30 2014

			992^4 + 992 + 1 (968381957089) and 992^4 + 992 - 1 (968381957087) are twin primes. Thus, 992 is a member of this sequence.

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

Intersection of A236759 and A049408.

[n: n in [1..5*10^3] |IsPrime(n^4+n-1) and IsPrime(n^4 +n+1)]; // Vincenzo Librandi, Dec 26 2015

Select[Range[3000], PrimeQ[#^4 + # - 1] && PrimeQ[#^4 + # + 1] &] (* Vincenzo Librandi, Dec 26 2015 *)
Select[Range[3000],AllTrue[#^4+#+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 13 2017 *)

s=[]; for(n=1, 3000, if(isprime(n^4+n+1)&&isprime(n^4+n-+1), s=concat(s, n))); s \\ Colin Barker, Jan 31 2014

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

A236950 Primes p such that p^4+p-1 is prime.

Original entry on oeis.org

2, 3, 13, 17, 23, 31, 43, 157, 229, 293, 353, 373, 397, 409, 463, 521, 577, 619, 727, 743, 857, 883, 953, 1049, 1087, 1123, 1171, 1319, 1409, 1423, 1429, 1459, 1499, 1511, 1543, 1619, 1693, 1847, 1871, 1931, 1951, 1993, 2017, 2029, 2129

Offset: 1

Derek Orr, Feb 01 2014

nonn

Primes in the sequence A236759.

			577 is prime and 577^4 + 577 - 1 = 110841719617 is prime. Thus, 577 is a member of this sequence.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A236759.

[p: p in PrimesUpTo(6000) | IsPrime(p^4+p-1)]; // Vincenzo Librandi, Feb 14 2014

Select[Prime[Range[5000]], PrimeQ[#^4 + # - 1]&] (* Vincenzo Librandi, Feb 14 2014 *)

s=[]; forprime(p=2, 6000, if(isprime(p^4+p-1), s=concat(s, p))); s \\ Colin Barker, Feb 05 2014

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

A236766 Numbers n such that n^4 +/- n +/- 1 are prime for all four possibilities.

Original entry on oeis.org

6, 9, 7266, 115131, 380529, 536646, 636609, 818526, 922734, 1389015, 1660119, 2045415, 2085726, 2262420, 2469396, 2722260, 2836374, 2954250, 3146904, 3614226, 3949770, 4062465, 4110834, 4211499, 4400100, 5081055, 5324424, 5434794, 5436090

Offset: 1

Derek Orr, Jan 30 2014

nonn

			380529^4+380529+1 (20967711831335262645811), 380529^4+380529-1 (20967711831335262645809), 380529^4-380529+1 (20967711831335261884753), and 380529^4-380529-1 (20967711831335261884751) are all prime. Thus, 380529 is a member of this sequence.

Intersection of A236759, A049408, A236761 and A126424.

Select[Range[55*10^5],AllTrue[#^4+{(#-1),(#+1),(-#-1),(-#+1)},PrimeQ]&] (* Harvey P. Dale, Nov 02 2024 *)

for(n=1, 5000000, if(isprime(n^4+n+1)&&isprime(n^4-n+1)&&isprime(n^4+n-1)&&isprime(n^4-n-1), print1(n, ","))) \\ Colin Barker, Jan 31 2014

import sympy
from sympy import isprime
{print(n) for n in range(10**7) if isprime(n**4+n+1) and isprime(n**4-n+1) and isprime(n**4+n-1) and isprime(n**4-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

A236760 Numbers n such that n^4 + n +- 1 are twin primes.

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A236950 Primes p such that p^4+p-1 is prime.

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A236766 Numbers n such that n^4 +/- n +/- 1 are prime for all four possibilities.

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Mathematica

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

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Mathematica

PARI