A007749 Numbers k such that k!! - 1 is prime.

3, 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, 728, 842, 888, 2328, 3326, 6404, 8670, 9682, 27056, 44318, 76190, 100654, 145706

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• nice

Comments

a(n) is even for n>1. a(n) = 2*A091415(n-1) for n>1, where A091415(n) = {2, 3, 4, 8, 13, 32, 41, 45, 59, 97, 107, 364, 421, 444, 1164, 1738, 3202, 4335, 4841, ...} (numbers k such that k!*2^k - 1 is prime). Corresponding primes of the form k!!-1 are listed in A117141 = {2, 7, 47, 383, 10321919, 51011754393599, ...}. - Alexander Adamchuk, Nov 19 2006

The PFGW program has been used to certify all the terms up to a(25), using a deterministic test which exploits the factorization of a(n) + 1. - Giovanni Resta, Apr 22 2016

References

• The Top Ten (a Catalogue of Primal Configurations) from the unpublished collections of R. Ondrejka, assisted by C. Caldwell and H. Dubner, March 11, 2000, Page 61.

Links

• Ken Davis, Status of Search for Multifactorial Primes.
• Ken Davis, Results for n!2-1.
• Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
• Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations
• Eric Weisstein's World of Mathematics, Double Factorial Prime
• Eric Weisstein's World of Mathematics, Integer Sequence Primes
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A006882.

Cf. A091415 (n such that n!*2^n - 1 is prime), A117141 (primes of the form n!! - 1).

Programs

Maple

select(t -> isprime(doublefactorial(t)-1), [3, seq(n,n=4..3000,2)]); # Robert Israel, Apr 21 2016

Mathematica

a(1) = 3, for n>1 k=2;f=2;Do[k=k+2;f=f*k;If[PrimeQ[f-1],Print[k]],{n,2,5000}] (* Alexander Adamchuk, Nov 19 2006 *)
Select[Range[45000],PrimeQ[#!!-1]&] (* Harvey P. Dale, Aug 07 2013 *)

PARI

print1(3);for(n=2, 1e3, if(ispseudoprime(n!<Charles R Greathouse IV, Jun 16 2011

Formula

a(n) = 2*A091415(n-1) for n>1. - Alexander Adamchuk, Nov 19 2006

Extensions

Entry updated by Robert G. Wilson v, Aug 18 2000

Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

a(23)-a(24) from Sou Fukui, Jun 05 2015

a(25) from Sou Fukui, Apr 21 2016