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URL: https://oeis.org/A173321

⇱ A173321 - OEIS

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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, 9731608032706559999, 204363768686837759999, 4496002911110430719999
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. (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
FORMULA
a(n) = n*a(n-1) + n - 1 for n > 0, a(0) = 3. - Vincenzo Librandi, Sep 30 2013
From Elmo R. Oliveira, Nov 01 2025: (Start)
a(n) = 4*A000142(n) - 1.
E.g.f.: 4/(1-x) - exp(x). (End)
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]]; // 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
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. A000142, A002145 (primes of the form 4*k+3), A333924.
Sequence in context: A232368 A333924 A383908 * A191498 A065747 A363398
KEYWORD
nonn,easy
AUTHOR
Vincenzo Librandi, Feb 16 2010
STATUS
approved