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%I #76 Mar 08 2023 03:39:16
%S 19,29,59,79,89,109,139,149,179,199,229,239,269,349,359,379,389,409,
%T 419,439,449,479,499,509,569,599,619,659,709,719,739,769,809,829,839,
%U 859,919,929,1009,1019,1039,1049,1069,1109,1129,1229,1249,1259,1279,1289
%N Primes of form 10*k + 9.
%C Also primes of form 5*k + 4.
%C 5 is quadratic residue of primes of form 10*k-1. - _Vincenzo Librandi_, Jun 25 2014
%C Also, primes p such that 5 divides sigma(p), cf. A274397. - _M. F. Hasler_, Jul 10 2016
%C 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
%C The digital root of a(n) is 1, 2, 4, 5, 7 or 8. - _Muniru A Asiru_, Apr 28 2018
%C From _Jianing Song_, Sep 13 2022: (Start)
%C 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]).
%C 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)
%H Michael B. Porter, <a href="/A030433/b030433.txt">Table of n, a(n) for n = 1..100000</a>
%H A. Granville and G. Martin, <a href="https://arxiv.org/abs/math/0408319">Prime number races</a>, arXiv:math/0408319 [math.NT], 2004.
%H Erika Klarreich, <a href="https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/">Mathematicians Discover Prime Conspiracy</a>, Quanta Magazine, 2016.
%H R. J. Lemke Oliver and K. Soundararajan, <a href="https://arxiv.org/abs/1603.03720">Unexpected biases in the distribution of consecutive primes</a>, arXiv:1603.03720 [math.NT], 2016.
%F a(n) = 10*A102700(n) + 9.
%F Union of A132234 and A132236. - _Ray Chandler_, Apr 07 2009
%F Intersection of A000040 and A017377. - _Iain Fox_, Dec 30 2017
%p select(isprime,[seq(10*n+9,n=1..500)]); # _Muniru A Asiru_, Apr 27 2018
%t Select[Prime@Range[210], Mod[ #, 10] == 9 &] (* _Ray Chandler_, Nov 07 2006 *)
%t Select[Range[9, 1300, 10], PrimeQ] (* _Harvey P. Dale_, Jun 01 2012 *)
%t Prime@Flatten@Position[Length@FactorList[((1+d)^5-1)/d,Modulus->#]&/@Prime@Range@200,3] (* _Federico Provvedi_, Apr 04 2018 *)
%o (PARI) select(n->n%10==9, primes(100)) \\ _Charles R Greathouse IV_, Apr 29 2015
%o (PARI) for(n=1, 1e3, if(isprime(p=10*n+9), print1(p, ", "))); \\ _Altug Alkan_, Apr 19 2018
%o (GAP) Filtered(List([1..500],n->10*n+9),IsPrime); # _Muniru A Asiru_, Apr 27 2018
%Y Cf. A045457, A049510, A102700.
%K nonn
%O 1,1
%A _Warut Roonguthai_
%E Extended by _Ray Chandler_, Nov 07 2006
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