A217759 Primes of the form 4k+3 generated recursively: a(1)=3, a(n)= Min{p; p is prime; Mod[p,4]=3; p|4Q^2-1}, where Q is the product of all previous terms in the sequence.

3, 7, 43, 19, 6863, 883, 23, 191, 2927, 205677423255820459, 11, 163, 227, 9127, 59, 31, 71, 131627, 2101324929412613521964366263134760336303, 127, 1302443, 4065403, 107, 2591, 21487, 223, 12823, 167, 53720906651, 5452254637117019, 39827899, 11719, 131

Offset: 1

Author

Daran Gill, Mar 23 2013

Keywords

• nonn

Comments

Contrast A057207, where all the factors are congruent to 1 (mod 4), here only one is guaranteed to be congruent to 3 (mod 4).

Examples

			a(10) is 205677423255820459 because it is the only prime factor congruent to 3 (mod 4) of 4*(3*7*43*19*6863*883*23*191*2927)^2-1 = 5*13*2088217*256085119729*205677423255820459. The four smaller factors are all congruent to 1 (mod 4).
		

References

• Dirichlet,P.G.L (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, p. 13.

Links

• Daran Gill, Table of n, a(n) for n = 1..51
• Mersenne Forum, Two new sequences

Crossrefs

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002145, A057204-A057208.