A219117 -id:A219117 - cp's OEIS frontend

A057683 Numbers k such that k^2 + k + 1, k^3 + k + 1 and k^4 + k + 1 are all prime.

1, 2, 5, 6, 12, 69, 77, 131, 162, 426, 701, 792, 1221, 1494, 1644, 1665, 2129, 2429, 2696, 3459, 3557, 3771, 4350, 4367, 5250, 5670, 6627, 7059, 7514, 7929, 8064, 9177, 9689, 10307, 10431, 11424, 13296, 13299, 13545, 14154, 14286, 14306, 15137

Offset: 1

Harvey P. Dale, Oct 20 2000

After a(0) = 1, k^5 + k + 1 is never prime. Proof: k^5 + k + 1 = (k^2 + k + 1)*(k^3 - k^2 + 1). - Jonathan Vos Post, Oct 17 2007, edited by Robert Israel, Aug 01 2016

For n > 1, no terms == 1 (mod 3) or == 3 (mod 5). - Robert Israel, Jul 31 2016

			5 is included because 5^2 + 5 + 1 = 31, 5^3 + 5 + 1 = 131 and 5^4 + 5 + 1 = 631 are all prime.

Reinhard Zumkeller and Robert Israel, Table of n, a(n) for n = 1..10000 (n = 1..100 from Reinhard Zumkeller).

Cf. A049407.

Cf. Subsequence of A219117; A010051.

a057683 n = a057683_list !! (n-1)
a057683_list = filter (all (== 1) . p) [1..] where
   p x = map (a010051 . (+ (x + 1)) . (x ^)) [2..4]
-- Reinhard Zumkeller, Nov 12 2012

[n: n in [0..20000]|IsPrime(n^2+n+1) and IsPrime(n^3+n+1) and IsPrime(n^4+n+1)] // Vincenzo Librandi, Dec 20 2010

select(n -> isprime(n^4+n+1) and isprime(n^3+n+1) and isprime(n^2+n+1), [$1..50000]); # Robert Israel, Jul 31 2016

Select[Range[16000],And@@PrimeQ/@(Table[n^i+n+1,{i,2,4}]/.n->#)&]  (* Harvey P. Dale, Mar 28 2011 *)

from sympy import isprime
A057683_list = [n for n in range(10**5) if isprime(n**2+n+1) and isprime(n**3+n+1) and isprime(n**4+n+1)] # Chai Wah Wu, Apr 02 2021

A219183 Numbers n such that n^1+n+1, n^2+n+1, n^3+n+1 and n^4+n+1 are all semiprime.

Original entry on oeis.org

84, 92, 129, 132, 182, 185, 195, 201, 234, 255, 264, 327, 333, 356, 407, 444, 449, 528, 705, 732, 794, 795, 881, 980, 1079, 1095, 1115, 1126, 1241, 1253, 1302, 1431, 1479, 1496, 1574, 1772, 1781, 1799, 1805, 1874, 1922, 2052, 2067, 2316, 2352, 2381, 2420

Offset: 1

Jonathan Vos Post, Nov 13 2012

nonn

This is to semiprimes A001358 what A219117 is to primes A000040. - Franklin T. Adams-Watters

From Robert Gerbicz: there is no n for which n^k+n+1 is semiprime for k=1,2,3,4,5. Proof: n^5+n+1 = (n^2+n+1)*(n^3-n^2+1), here n^2+n+1 is semiprime, so for n > 1, n^5+n+1 has at least 3 factors, hence not a semiprime.

			a(1) = 84 because 84^4 + 84 + 1 = 49787221 = 11 * 4526111; 84^3 + 84 + 1 = 592789 = 29 * 20441; 84^2 + 84 + 1 = 7141 = 37 * 193; 84^1 + 84 + 1 = 169 = 13^2.
3^4+3+1 = 85 = 5*17 is semiprime, but 3^3+3+1 = 321 is prime, so 3 is not in this sequence.
8^4+8+1 = 4105 = 5 * 821 is semiprime, but 8^3+8+1 = 521 is prime, so 8 is not in this sequence.
20^4+20+1 = 160021 = 17 * 9413 is semiprime, and 20^3+20+1 = 8021 = 13 * 617 is semiprime, but 20^2+20+1 = 421 is prime, so 20 is not in this sequence.

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

Cf. A001358, A219117.

s:=func; [k : k in [2..2500]| forall{i:i in [1,2,3,4]| s(k^i+k+1)}]; // Marius A. Burtea, Feb 11 2020

is(n)=vector(4,i,bigomega(n^i+n+1))==[2,2,2,2] \\ Charles R Greathouse IV, Nov 13 2012

A236045 Primes p such that p^1+p+1, p^2+p+1, p^3+p+1, and p^4+p+1 are all prime.

Original entry on oeis.org

2, 5, 131, 2129, 9689, 27809, 36821, 46619, 611729, 746171, 987491, 1121189, 1486451, 2215529, 2701931, 4202171, 4481069, 4846469, 5162141, 5605949, 6931559, 7181039, 8608571, 9276821, 9762611, 11427491, 11447759, 12208019

Offset: 1

Derek Orr, Jan 18 2014

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..150

Cf. A219117.

Select[Prime[Range[810000]],And@@PrimeQ[Table[#^n+#+1,{n,4}]]&] (* Harvey P. Dale, Apr 07 2014 *)

list(maxx)={n=2; cnt=0; while(nBill McEachen, Feb 05 2014

import sympy
from sympy import isprime
{print(p) for p in range(10**8) if isprime(p) and isprime(p**1+p+1) and isprime(p**2+p+1) and isprime(p**3+p+1) and isprime(p**4+p+1)}

cp's OEIS Frontend

A057683 Numbers k such that k^2 + k + 1, k^3 + k + 1 and k^4 + k + 1 are all prime.

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A219183 Numbers n such that n^1+n+1, n^2+n+1, n^3+n+1 and n^4+n+1 are all semiprime.

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A236045 Primes p such that p^1+p+1, p^2+p+1, p^3+p+1, and p^4+p+1 are all prime.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python