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

1,1

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

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A178319 A202943 A355088 * A124988 A220934 A221288

Adjacent sequences: A057201 A057202 A057203 * A057205 A057206 A057207

Keyword

nonn

Author

Labos Elemer, Oct 09 2000

Extensions

More terms from Nick Hobson, Nov 14 2006

More terms from Sean A. Irvine, Oct 23 2014

Status

approved