A190527 Primes of the form p^4 + p^3 + p^2 + p + 1, where p is prime.

31, 2801, 30941, 88741, 292561, 732541, 3500201, 28792661, 39449441, 48037081, 262209281, 1394714501, 2666986681, 3276517921, 4802611441, 5908670381, 12936304421, 16656709681, 19408913261, 24903325661, 37226181521, 43713558101, 52753304641, 64141071121, 96427561501, 100648118041

Offset: 1

Author

Bernard Schott, Dec 20 2012

Keywords

• nonn

Comments

These primes are generated by exactly A065509, cf. 2nd formula.

These numbers are repunit primes 11111_p, so they are Brazilian primes (A085104).

When p^4 + p^3 + p^2 + p + 1 = sigma(p^4) is prime, then it equals A193574(p^4), so that this sequence is a subsequence of A193574; by definition it is also a subsequence of A053699 and A131992. - Hartmut F. W. Hoft, May 05 2017

Examples

			a(3) = 30941 = 11111_13 = 13^4 + 13^3 + 13^2 + 13^1 + 1 is prime.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1100

Crossrefs

Cf. A049409 (n^4 + ... + 1 is prime), A065509 (primes among these n), A193574.

Subsequence of A088548 (primes n^4 + ... + 1) and A085104 ("Brazilian" primes, of the form 1 + n + n^2 + ... + n^k).

Intersection of A000040 (primes) and A131992 (p^4 + ... + 1), subsequence of A053699 (n^4 + ... + 1).

Programs

Magma

[p: p in PrimesUpTo(600) | IsPrime(p) where p is p^4 +p^3+p^2+p+1]; // Vincenzo Librandi, May 06 2017

Mathematica

a190527[n_] := Select[Map[(Prime[#]^5-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a190527[100] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[#^4 + #^3 + #^2 + # + 1 &/@Prime[Range[100]], PrimeQ] (* Vincenzo Librandi, May 06 2017 *)

PARI

[q|p<-primes(100),ispseudoprime(q=(p^5-1)\(p-1))]
A190527_vec(N)=[(p^5-1)\(p-1)|p<-A065509_vec(N)] \\ M. F. Hasler, Mar 03 2020

Formula

a(n) = A193574(A065509(n)^4). - Hartmut F. W. Hoft, May 08 2017

a(n) = A053699(A065509(n)) = A000203(A065509(n)^4). - M. F. Hasler, Mar 03 2020

Extensions

a(7) corrected and a(18)-a(26) added by Hartmut F. W. Hoft, May 05 2017

Edited by M. F. Hasler, Mar 06 2020