A030433 Primes of form 10*k + 9.

19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, 509, 569, 599, 619, 659, 709, 719, 739, 769, 809, 829, 839, 859, 919, 929, 1009, 1019, 1039, 1049, 1069, 1109, 1129, 1229, 1249, 1259, 1279, 1289

Offset: 1

Author

Warut Roonguthai

Keywords

• nonn

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.

Crossrefs

Cf. A045457, A049510, A102700.

Programs

GAP

Filtered(List([1..500],n->10*n+9),IsPrime); # Muniru A Asiru, Apr 27 2018

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 *)

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

Formula

a(n) = 10*A102700(n) + 9.

Union of A132234 and A132236. - Ray Chandler, Apr 07 2009

Intersection of A000040 and A017377. - Iain Fox, Dec 30 2017

Extensions

Extended by Ray Chandler, Nov 07 2006