A057204 Primes congruent to 1 mod 6 generated recursively. Initial prime is 7. The next term is p(n) = Min_{p is prime; p divides 4Q^2+3; p mod 6 = 1}, where Q is the product of previous entries of the sequence.

7, 199, 7761799, 487, 67, 103, 3562539697, 7251847, 13, 127, 5115369871402405003, 31, 697830431171707, 151, 3061, 229, 193, 5393552285540920774057256555028583857599359699, 709, 397, 37, 61, 46168741, 3127279, 181, 122268541

Offset: 1

Author

Labos Elemer, Oct 09 2000

Keywords

• nonn

Comments

4*Q^2 + 3 always has a prime divisor congruent to 1 modulo 6.

If we start with the empty product Q=1 then it is not necessary to specify the initial prime. - Jens Kruse Andersen, Jun 30 2014

Examples

			a(4)=487 is the smallest prime divisor of 4*Q*Q + 3 = 10812186007, congruent to 1 (mod 6), where Q = 7*199*7761799.
		

References

• P. G. L. Dirichlet (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

• Sean A. Irvine, Table of n, a(n) for n = 1..48

Crossrefs

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002476, A057204-A057208.

Programs

Mathematica

a={7}; q=1;
For[n=2,n<=7,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4*q^2+3][[All,1]],Mod[#,6]==1 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

PARI

Q=1;for(n=1,11,f=factor(4*Q^2+3);for(i=1,#f~,p=f[i,1];if(p%6==1,break));print1(p", ");Q*=p) \\ Jens Kruse Andersen, Jun 30 2014

Extensions

More terms from Nick Hobson, Nov 14 2006

More terms from Sean A. Irvine, Oct 23 2014