A057208 Primes of the form 8k+5 generated recursively: a(1)=5, a(n) = least prime p == 5 (mod 8) with p | 4+Q^2, where Q is the product of all previous terms in the sequence.

5, 29, 1237, 32171803229, 829, 405565189, 14717, 39405395843265000967254638989319923697097319108505264560061, 282860648026692294583447078797184988636062145943222437, 53, 421, 13, 109, 4133, 6476791289161646286812333, 461, 34549, 453690033695798389561735541

Offset: 1

Author

Labos Elemer, Oct 09 2000

Keywords

• nonn

Examples

			a(3) = 1237 = 8*154 + 5 is the smallest suitable prime divisor of (5*29)*5*29 + 4 = 21029 = 17*1237. (Although 17 is the smallest prime divisor, 17 is not congruent to 5 modulo 8.)
		

References

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

Links

• P. G. L. Dirichlet, Supplement VI: Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, 1871, 24 pages.

Crossrefs

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A007521, A057204-A057208.

Programs

Mathematica

a={5}; q=1;
For[n=2,n<=7,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[4+q^2][[All,1]],Mod[#,8]==5 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

PARI

lista(nn) = {v = vector(nn); v[1] = 5; print1(v[1], ", "); for (n=2, nn, f = factor(4 + prod(k=1, n-1, v[k])^2); for (k=1, #f~, if (f[k, 1] % 8 == 5, v[n] = f[k,1]; break);); print1(v[n], ", "););} \\ Michel Marcus, Oct 27 2014

Extensions

More terms from Sean A. Irvine, Oct 26 2014