A057205 Primes congruent to 3 modulo 4 generated recursively: a(n) = Min_{p, prime; p mod 4 = 3; p|4Q-1}, where Q is the product of all previous terms in the sequence. The initial term is 3.

3, 11, 131, 17291, 298995971, 8779, 594359, 59, 151, 983, 19, 38851089348584904271503421339, 2359886893253830912337243172544609142020402559023, 823818731, 2287, 7, 9680188101680097499940803368598534875039120224550520256994576755856639426217960921548886589841784188388581120523, 163, 83, 1471, 34211, 2350509754734287, 23567

Offset: 1

Author

Labos Elemer, Oct 09 2000

Keywords

• nonn

Examples

			a(4) = 17291 = 4*4322 + 3 is the smallest prime divisor congruent to 3 (mod 4) of Q = 3*11*131 - 1 = 17291.
		

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.

Crossrefs

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

Programs

Mathematica

a={3}; q=1;
For[n=2,n<=7,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4*q-1][[All,1]],Mod[#,4]==3&]]];
    ];
a (* Robert Price, Jul 18 2015 *)

Extensions

More terms from Phil Carmody, Sep 18 2005

Terms corrected and extended by Sean A. Irvine, Oct 23 2014