A057206 Primes of the form 6k+5 generated recursively: a(1)=5; a(n) = min{p, prime; p mod 6 = 5; p | 6Q-1}, where Q is the product of all previous terms in the sequence.

5, 29, 11, 1367, 13082189, 89, 59, 29819952677, 91736008068017, 17, 887050405736870123700827, 688273423680369013308306870159348033807942418302818522537, 74367405177105011, 12731422703, 1812053

Offset

1,1

Comments

There are infinitely many primes of the form 6k + 5, and this sequence figures in the classic proof of that fact. - Alonso del Arte, Mar 02 2017

References

Dirichlet, P. G. L. (1871): Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.

Links

Robert Price, Table of n, a(n) for n = 1..17

Example

a(3) = 11 is the smallest prime divisor of the form 6k + 5 of 6 * (5 * 29) - 1 = 6Q - 1 = 11 * 79 = 869.

Mathematica

primes5mod6 = {5}; q = 1; For[n = 2, n <= 10, n++, q = q * Last[ primes5mod6]; AppendTo[primes5mod6, Min[Select[FactorInteger[6 * q - 1][[All, 1]], Mod[#, 6] == 5 &]]]; ]; primes5mod6 (* Robert Price, Jul 18 2015 *)

Prog

(PARI) main(size)={my(v=vector(size), i, q=1, t); for(i=1, size, t=1; while(!(prime(t)%6==5&&(6*q-1)%prime(t)==0), t++); v[i]=prime(t); q*=v[i]); v; } /* Anders Hellström, Jul 18 2015 */

Crossrefs

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A057204-A057208, A007528.

Sequence in context: A083020 A033503 A181616 * A057713 A124987 A002584

Adjacent sequences: A057203 A057204 A057205 * A057207 A057208 A057209

Keyword

nonn

Author

Labos Elemer, Oct 09 2000

Extensions

a(13)-a(17) from Robert Price, Jul 18 2015

Status

approved