A333924 Smallest prime of the form 4*k + 3 that is a divisor of 4*n! - 1.

3, 3, 7, 23, 19, 479, 2879, 19, 179, 2551, 14515199, 159667199, 26246663, 47, 3007159, 85303, 43, 455999, 13099, 311369011223, 7791519641878751, 59, 50207, 149709500816123, 71, 61651424911, 1146111319366855507, 3902575987, 27963070149883187169101323, 3262754470190705587633531

Offset: 0

Author

Bernard Schott, Apr 10 2020

Keywords

• nonn

Comments

Every integer equal to 4*n!-1 (A173321) has a prime factor > n of the form 4*k+3; this is one of the proofs which show that there are infinitely many primes of the form 4*k+3 (A002145).

Examples

			4*11!-1 = 159667199 that is prime of the form 4*k+3, hence a(11) = 159667199.
4*13!-1 = 24908083199 = 47 * 2963 * 178859, these 3 prime factors are all of the form 4*k+3, the smallest one is 47 hence a(13) = 47.
4*14!-1 = 348713164799 = 61 * 1901 * 3007159, only 3007159 is a prime of the form 4*k+3, hence a(14) = 3007159.
		

Crossrefs

Cf. A002144, A173321.

Subsequence of A002145.

Programs

Mathematica

a[n_] := Min[Select[First /@ FactorInteger[4*n! - 1], Mod[#, 4] == 3 &]]; Array[a, 30, 0] (* Amiram Eldar, Apr 10 2020 *)

PARI

a(n) = {my(f=factor(4*n!-1)[,1]); for(i=1, #f, if(f[i]%4==3, return(f[i]))); } \\ Jinyuan Wang, Apr 10 2020

Extensions

a(23) corrected by and more terms from Jinyuan Wang, Apr 10 2020