login
a(n) = 4*n! - 1.
7

%I #35 Sep 08 2022 08:45:50

%S 3,3,7,23,95,479,2879,20159,161279,1451519,14515199,159667199,

%T 1916006399,24908083199,348713164799,5230697471999,83691159551999,

%U 1422749712383999,25609494822911999,486580401635327999

%N a(n) = 4*n! - 1.

%C From _Bernard Schott_, Jul 11 2019: (Start)

%C With this sequence, it is possible to prove that there are infinitely many prime numbers of the form 4*k+3.

%C Prove that:

%C 1. Every prime factor of a(n) is > n, and,

%C 2. All these prime factors are of the form 4*k+1 or 4*k+3.

%C 3. There is at least one prime of the form 4*k+3 > n,

%C 4. The set of prime numbers of the form 4*k+3 is infinite.

%C (End)

%C The smallest prime of the form 4*k + 3 that divides a(n) is A333924(n). - _Bernard Schott_, Oct 08 2021

%D Transmath, Term S, Spécialité, Programme 2002, Nathan, 2002, Exercice 82 p. 93.

%H Vincenzo Librandi, <a href="/A173321/b173321.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = n*a(n-1) + n - 1 for n > 0, a(0) = 3. - _Vincenzo Librandi_, Sep 30 2013

%p A173321:=n->4*n! - 1; seq(A173321(n), n=0..25); # _Wesley Ivan Hurt_, Jan 24 2014

%t Table[4 n! - 1, {n, 0, 25}] (* _Vincenzo Librandi_, Sep 30 2013 *)

%o (Magma) [4*Factorial(n)-1: n in [0..25]]; /* or */ [3] cat [n eq 1 select n+2 else n*Self(n-1)+n-1: n in [1..25] ]; // _Vincenzo Librandi_, Sep 30 2013

%Y Cf. sequences of the type k*n!-1: A033312 (k=1), A020543 (k=2), A173323 (k=3), this sequence, A173317 (k=5), A173316 (k=6).

%Y Cf. A002145 (primes of the form 4*k+3), A333924.

%K nonn,easy

%O 0,1

%A _Vincenzo Librandi_, Feb 16 2010