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%I #57 Sep 08 2022 08:45:57
%S 31,2801,30941,88741,292561,732541,3500201,28792661,39449441,48037081,
%T 262209281,1394714501,2666986681,3276517921,4802611441,5908670381,
%U 12936304421,16656709681,19408913261,24903325661,37226181521,43713558101,52753304641,64141071121,96427561501,100648118041
%N Primes of the form p^4 + p^3 + p^2 + p + 1, where p is prime.
%C These primes are generated by exactly A065509, cf. 2nd formula.
%C These numbers are repunit primes 11111_p, so they are Brazilian primes (A085104).
%C 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
%H Vincenzo Librandi, <a href="/A190527/b190527.txt">Table of n, a(n) for n = 1..1100</a>
%F a(n) = A193574(A065509(n)^4). - _Hartmut F. W. Hoft_, May 08 2017
%F a(n) = A053699(A065509(n)) = A000203(A065509(n)^4). - _M. F. Hasler_, Mar 03 2020
%e a(3) = 30941 = 11111_13 = 13^4 + 13^3 + 13^2 + 13^1 + 1 is prime.
%t a190527[n_] := Select[Map[(Prime[#]^5-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
%t a190527[100] (* data *) (* _Hartmut F. W. Hoft_, May 05 2017 *)
%t Select[#^4 + #^3 + #^2 + # + 1 &/@Prime[Range[100]], PrimeQ] (* _Vincenzo Librandi_, May 06 2017 *)
%o (Magma)[p: p in PrimesUpTo(600) | IsPrime(p) where p is p^4 +p^3+p^2+p+1]; // _Vincenzo Librandi_, May 06 2017
%o (PARI)
%o [q|p<-primes(100),ispseudoprime(q=(p^5-1)\(p-1))]
%o A190527_vec(N)=[(p^5-1)\(p-1)|p<-A065509_vec(N)] \\ _M. F. Hasler_, Mar 03 2020
%Y Cf. A049409 (n^4 + ... + 1 is prime), A065509 (primes among these n), A193574.
%Y Subsequence of A088548 (primes n^4 + ... + 1) and A085104 ("Brazilian" primes, of the form 1 + n + n^2 + ... + n^k).
%Y Intersection of A000040 (primes) and A131992 (p^4 + ... + 1), subsequence of A053699 (n^4 + ... + 1).
%K nonn
%O 1,1
%A _Bernard Schott_, Dec 20 2012
%E a(7) corrected and a(18)-a(26) added by _Hartmut F. W. Hoft_, May 05 2017
%E Edited by _M. F. Hasler_, Mar 06 2020
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