A049409 Numbers n such that n^4 + n^3 + n^2 + n + 1 is prime.

1, 2, 7, 12, 13, 17, 22, 23, 24, 28, 29, 30, 40, 43, 44, 50, 62, 63, 68, 73, 74, 77, 79, 83, 85, 94, 99, 110, 117, 118, 120, 122, 127, 129, 134, 143, 145, 154, 162, 164, 165, 172, 175, 177, 193, 198, 204, 208, 222, 227, 239, 249, 254, 255, 260

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

There is no square > 1 in this sequence, because if f(n) = n^4 + n^3 + n^2 + n + 1, then f(n^2) = f(n)*f(-n). Actually, f(x) divides f(x^m) for all m not in 5Z. So the only perfect powers in this sequence can be 5th, 25th, 125th... powers. The least perfect power > 1 in this sequence is 22^5. - M. F. Hasler, Feb 09 2012

The corresponding prime numbers n^4 + n^3 + n^2 + n + 1 are in A088548. - Bernard Schott, Dec 19 2012

This is also the list of bases where 11111 is a prime number. - Christian N. K. Anderson, Mar 28 2013

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A088548.

Programs

Maple

A049409:=n->if isprime(n^4+n^3+n^2+n+1) then n fi; seq(A049409(n), n=1..300); # Wesley Ivan Hurt, Dec 28 2013

Mathematica

lst={}; Do[p=n^0+n^1+n^2+n^3+n^4; If[PrimeQ[p], AppendTo[lst,n]], {n,300}]; lst (* Vladimir Joseph Stephan Orlovsky, Jun 10 2009 *)
Select[Range[300],PrimeQ[1+Total[#^Range[4]]]&] (* Harvey P. Dale, Mar 12 2018 *)

PARI

for(n=1,1000, ispseudoprime(n^4+n^3+n^2+n+1) & print1(n","))  \\ M. F. Hasler, Feb 09 2012