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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

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OFFSET

1,1

REFERENCES

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

LINKS

Table of n, a(n) for n=1..18.

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.

EXAMPLE

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.)

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A273918 A383741 A085553 * A175905 A057706 A057705

Adjacent sequences: A057205 A057206 A057207 * A057209 A057210 A057211

KEYWORD

nonn

AUTHOR

Labos Elemer, Oct 09 2000

EXTENSIONS

More terms from Sean A. Irvine, Oct 26 2014

STATUS

approved

URL: https://oeis.org/A057208

⇱ A057208 - OEIS