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

%I A007749 #63 Mar 19 2025 08:24:52
%S A007749 3,4,6,8,16,26,64,82,90,118,194,214,728,842,888,2328,3326,6404,8670,
%T A007749 9682,27056,44318,76190,100654,145706
%N A007749 Numbers k such that k!! - 1 is prime.
%C A007749 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
%C A007749 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
%D A007749 The Top Ten (a Catalogue of Primal Configurations) from the unpublished collections of R. Ondrejka, assisted by C. Caldwell and H. Dubner, March 11, 2000, Page 61.
%H A007749 Ken Davis, <a href="http://mfprimes.free-dc.org">Status of Search for Multifactorial Primes.</a>
%H A007749 Ken Davis, <a href="http://mfprimes.free-dc.org/mfdata/f02m.html">Results for n!2-1.</a>
%H A007749 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_16">Fascinating Factorials</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
%H A007749 Rudolf Ondrejka, <a href="https://t5k.org/lists/top_ten/">The Top Ten: a Catalogue of Primal Configurations</a>
%H A007749 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleFactorialPrime.html">Double Factorial Prime</a>
%H A007749 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>
%H A007749 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A007749 a(n) = 2*A091415(n-1) for n>1. - _Alexander Adamchuk_, Nov 19 2006
%p A007749 select(t -> isprime(doublefactorial(t)-1), [3, seq(n,n=4..3000,2)]); # _Robert Israel_, Apr 21 2016
%t A007749 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 *)
%t A007749 Select[Range[45000],PrimeQ[#!!-1]&] (* _Harvey P. Dale_, Aug 07 2013 *)
%o A007749 (PARI) print1(3);for(n=2, 1e3, if(ispseudoprime(n!<<n-1), print1(", ",2*n))) \\ _Charles R Greathouse IV_, Jun 16 2011
%Y A007749 Cf. A006882.
%Y A007749 Cf. A091415 (n such that n!*2^n - 1 is prime), A117141 (primes of the form n!! - 1).
%K A007749 nonn,hard,nice
%O A007749 1,1
%A A007749 _N. J. A. Sloane_
%E A007749 Entry updated by _Robert G. Wilson v_, Aug 18 2000
%E A007749 Corrected and extended by Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
%E A007749 a(23)-a(24) from _Sou Fukui_, Jun 05 2015
%E A007749 a(25) from _Sou Fukui_, Apr 21 2016