Daran Gill - cp's OEIS frontend

User: Daran Gill

Daran Gill has authored 2 sequences.

A218467 A variant of the Euclid-Mullin sequence A000945: a(1) = 2, a(n+1) is smallest prime factor congruent to 3 (mod 4) of Product_{k=1..n} a(k) + 1.

2, 3, 7, 43, 139, 50207, 23, 10651, 563, 11, 19, 363303615453958067659, 787, 2803, 3261639461817858097484047657974700766171, 448513341328399688966874038187266281752082128599801650127, 89724193529143

Offset: 1

Daran Gill, Mar 26 2013

nonn

Just as the Euclid-Mullin sequence is suggested by Euclid's proof of an infinity of primes, this sequence is suggested by a variation of his proof, showing the existence of an infinity of primes congruent to 3 (mod 4).  See Hardy and Wright in the Reference below.
Could also be viewed as a variation on A217759.  Restricting the scope of "smallest prime factor congruent to 3 (mod 4)" to the larger of the two algebraic factors of 4Q^2-1 as defined in that sequence results in a sequence essentially the same as this one.
a(18) has 149 digits.

			This sequence and A000945 are identical up to their fourth term.  The fifth terms of both that sequence and this one are factors of 2*3*7*43+1=13*139.  The smallest factor, used by A000945, is congruent to 1 (mod 4).  Here we take the larger.

P. G. L. Dirichlet: Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, (1871), 24 pages.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 13.

Daran Gill, Table of n, a(n) for n = 1..19

a={2}; q=1;
For[n=2,n<=12,n++,
    q=q*Last[a];
    AppendTo[a,Min[Select[FactorInteger[q+1][[All,1]],Mod[#,4]==3 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

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.

Original entry on oeis.org

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

Daran Gill, Mar 23 2013

nonn

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

			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).

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.

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

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

cp's OEIS Frontend

User: Daran Gill

A218467 A variant of the Euclid-Mullin sequence A000945: a(1) = 2, a(n+1) is smallest prime factor congruent to 3 (mod 4) of Product_{k=1..n} a(k) + 1.

Views

Author

Keywords

Comments

Examples

References

Links

Programs

Mathematica