%I #27 Sep 08 2022 08:45:12

%S 5,31,2801,22621,30941,88741,245411,292561,346201,637421,732541,

%T 837931,2625641,3500201,3835261,6377551,15018571,16007041,21700501,

%U 28792661,30397351,35615581,39449441,48037081,52822061,78914411,97039801,147753211,189004141,195534851

%N Primes of the form k^4 + k^3 + k^2 + k + 1.

%C These numbers when >= 31 are primes repunits 11111_n in a base n >= 2, so except 5, they are all Brazilian primes belonging to A085104. (See Links "Les nombres brésiliens", § V.4 - § V.5.) A008858 is generated by the bases n present in A049409. - _Bernard Schott_, Dec 19 2012

%H Vincenzo Librandi, <a href="/A088548/b088548.txt">Table of n, a(n) for n = 1..1000</a>

%H Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

%F A000040 intersect A053699. - _R. J. Mathar_, Feb 07 2014

%e a(2) = 31 is prime and 31 = 2^4 + 2^3 + 2^2 + 2 + 1.

%t lst={}; Do[a=1+n+n^2+n^3+n^4; If[PrimeQ[a], AppendTo[lst,a]], {n,6!}]; lst (* _Vladimir Joseph Stephan Orlovsky_, Jun 02 2009 *)

%t Select[Table[n^4+n^3+n^2+n+1, {n,0,2000}], PrimeQ] (* _Vincenzo Librandi_, Jul 16 2012 *)

%o (PARI) polypn(n,p) = { for(x=1,n, if(p%2,y=2,y=1); for(m=1,p, y=y+x^m; ); if(isprime(y),print1(y",")); ) }

%o (Magma) [a: n in [0..200] | IsPrime(a) where a is n^4+n^3+n^2+n+1]; // _Vincenzo Librandi_, Jul 16 2012

%o (Python)

%o from sympy import isprime

%o print(list(filter(isprime, (k**4+k**3+k**2+k+1 for k in range(120))))) # _Michael S. Branicky_, May 31 2021

%Y Cf. A002383, A049409, A085104, A088550.

%K nonn,easy

%O 1,1

%A _Cino Hilliard_, Nov 17 2003