A124985 Primes of the form 8*k + 7 generated recursively. Initial prime is 7. General term is a(n) = Min_{p is prime; p divides 8*Q^2 - 1; p == 7 (mod 8)}, where Q is the product of the previous terms.

7, 23, 207367, 1902391, 167, 1511, 28031, 79, 3142977463, 2473230126937097422987916357409859838765327, 2499581669222318172005765848188928913768594409919797075052820591, 223

Offset: 1

Author

Nick Hobson, Nov 18 2006

Keywords

• nonn

Comments

8*Q^2 - 1 always has a prime divisor congruent to 7 modulo 8.

Examples

			a(4) = 1902391 is the smallest prime divisor, congruent to 7 modulo 8, of 8*Q^2 - 1 = 8917046441372551 = 97 * 1902391 * 48322513, where Q = 7 * 23 * 207367.
		

References

• D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 182.

Crossrefs

Cf. A000945, A007522, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

Programs

Mathematica

a={7}; q=1;
For[n=2,n<=9,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[8*q^2-1][[All,1]],Mod[#,8]==7&]]];
    ];
a (* Robert Price, Jul 18 2015 *)

PARI

main(size)={my(v=vector(size),i,q=1,t);for(i=1,size,t=1;while(!(prime(t)%8==7&&(8*q^2-1)%prime(t)==0),t++);v[i]=prime(t);q*=v[i]);v;} /* Anders Hellström, Jul 18 2015 */

Extensions

Edited and added a(11)-a(12) by Max Alekseyev, May 31 2013