A173321 a(n) = 4*n! - 1.

3, 3, 7, 23, 95, 479, 2879, 20159, 161279, 1451519, 14515199, 159667199, 1916006399, 24908083199, 348713164799, 5230697471999, 83691159551999, 1422749712383999, 25609494822911999, 486580401635327999

Offset: 0

Author

Vincenzo Librandi, Feb 16 2010

Keywords

• nonn
• easy

Comments

From Bernard Schott, Jul 11 2019: (Start)

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

Prove that:

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

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

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

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

(End)

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

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

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).

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

Programs

Magma

[4*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013

Magma

[3] cat [n eq 1 select n+2 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013

Maple

A173321:=n->4*n! - 1; seq(A173321(n), n=0..25); # Wesley Ivan Hurt, Jan 24 2014

Mathematica

Table[4 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

Formula

a(n) = n*a(n-1) + n - 1 for n > 0, a(0) = 3. - Vincenzo Librandi, Sep 30 2013