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A173321 a(n) = 4*n! - 1. 7
3, 3, 7, 23, 95, 479, 2879, 20159, 161279, 1451519, 14515199, 159667199, 1916006399, 24908083199, 348713164799, 5230697471999, 83691159551999, 1422749712383999, 25609494822911999, 486580401635327999 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

Indeed, for n >= 1, the successive prime factors of a(n) that are of the form 4*k+3 are 3, 7, 23, 19, 479, 2879, 19, 179, ... (End)

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

FORMULA

a(n) = n*a(n-1) + n - 1 for n > 0, a(0) = 3. - 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 *)

PROG

(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

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).

Sequence in context: A262375 A232368 A333924 * A191498 A065747 A232309

Adjacent sequences:  A173318 A173319 A173320 * A173322 A173323 A173324

KEYWORD

nonn,easy

AUTHOR

Vincenzo Librandi, Feb 16 2010

STATUS

approved

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Last modified September 23 08:47 EDT 2021. Contains 347611 sequences. (Running on oeis4.)