OFFSET
1,1
COMMENTS
Also primes of form 5*k + 4.
5 is quadratic residue of primes of form 10*k-1. - Vincenzo Librandi, Jun 25 2014
Also, primes p such that 5 divides sigma(p), cf. A274397. - M. F. Hasler, Jul 10 2016
Conjecture: Primes p such that ((x+1)^5-1)/x has 2 distinct irreducible factors of degree 2 over GF(p). - Federico Provvedi, Apr 01 2018
The digital root of a(n) is 1, 2, 4, 5, 7 or 8. - Muniru A Asiru, Apr 28 2018
From Jianing Song, Sep 13 2022: (Start)
Primes p such that the ideal (p) factors into two prime ideals in Z[zeta_5], where zeta_5 = exp(2*Pi*i/5). Since Z[zeta_5] is a PID, this is equivalent to saying that this sequence lists primes p that are the product of two non-associate prime elements Z[zeta_5]. In particular, the factorization of p == 4 (mod 5) in Z[zeta_5] coincides with the factorization in Z[(1+sqrt(5))/2] (e.g., 19 = (8+3*sqrt(5))*(8-3*sqrt(5)) is the factorization of 19 in both Z[(1+sqrt(5))/2] and Z[zeta_5]).
Also primes p such that x^4 + x^3 + x^2 + x + 1 factors into two irreducible quadratic polynomials over GF(p) (cf. A327753). (End)
LINKS
Michael B. Porter, Table of n, a(n) for n = 1..100000
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Erika Klarreich, Mathematicians Discover Prime Conspiracy, Quanta Magazine, 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
FORMULA
MAPLE
select(isprime, [seq(10*n+9, n=1..500)]); # Muniru A Asiru, Apr 27 2018
MATHEMATICA
Select[Prime@Range[210], Mod[ #, 10] == 9 &] (* Ray Chandler, Nov 07 2006 *)
Select[Range[9, 1300, 10], PrimeQ] (* Harvey P. Dale, Jun 01 2012 *)
Prime@Flatten@Position[Length@FactorList[((1+d)^5-1)/d, Modulus->#]&/@Prime@Range@200, 3] (* Federico Provvedi, Apr 04 2018 *)
PROG
(PARI) select(n->n%10==9, primes(100)) \\ Charles R Greathouse IV, Apr 29 2015
(PARI) for(n=1, 1e3, if(isprime(p=10*n+9), print1(p, ", "))); \\ Altug Alkan, Apr 19 2018
(GAP) Filtered(List([1..500], n->10*n+9), IsPrime); # Muniru A Asiru, Apr 27 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Extended by Ray Chandler, Nov 07 2006
STATUS
approved