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, 27201015740779, 1319, 47, 1583, 179, 311, 15971

Offset

1,1

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).

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..40 [terms 30..40 corrected by Sean A. Irvine]

Mersenne Forum, Two new sequences

Example

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).

Crossrefs

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

Sequence in context: A229941 A019018 A018993 * A160615 A106965 A337829

Adjacent sequences: A217756 A217757 A217758 * A217760 A217761 A217762

Keyword

nonn

Author

Daran Gill, Mar 23 2013

Extensions

a(30) onward corrected by Sean A. Irvine, Feb 14 2026

Status

approved