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A084438 Positive integers k such that k!!! - 1 = A007661(k) - 1 is prime.

%I A084438 #29 Mar 19 2025 10:26:16
%S A084438 3,4,6,8,20,26,36,50,60,114,135,138,248,315,351,429,642,5505,8793,
%T A084438 12086,13580,23109,34626,34706,56282,57675,58298
%N A084438 Positive integers k such that k!!! - 1 = A007661(k) - 1 is prime.
%C A084438 The search for multifactorial primes started by Ray Ballinger is now continued by a team of volunteers on the website of Ken Davis (see link).
%H A084438 Ken Davis, <a href="http://mfprimes.free-dc.org">Status of Search for Multifactorial Primes</a>.
%H A084438 Ken Davis, <a href="http://mfprimes.free-dc.org/mfdata/f03m.html">Results for n!3-1</a>.
%H A084438 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 A084438 ProthSearch.net (on web.archive.org), <a href="http://web.archive.org/web/20050220234200/http://www.prothsearch.net/n3minus.txt">n3minus.txt</a>.
%e A084438 a(4) = 8 since 8!!! - 1 = 8*5*2 - 1 = 79 is the 4th prime of that form.
%e A084438 26!!! - 1 = 2504902399 is prime.
%t A084438 multiFactorial[n_, k_] := If[n < 1, 1, n * multiFactorial[n - k, k]];
%t A084438 Select[Range[0, 1000], PrimeQ[multiFactorial[#, 3] - 1] & ] (* _Robert Price_, Apr 19 2019 *)
%t A084438 Select[Range[650], PrimeQ[Times @@ Range[#, 1, -3] - 1] &] (* The program generates the first 17 terms of the sequence. To generate more, change the Range constant but the program may take a long time to run. *) (* _Harvey P. Dale_, May 22 2021 *)
%o A084438 (PARI) A007661(n) = prod(i=1,(n-1)\3,n-=3,n+!n)
%o A084438 for(n=1,999,if(isprime(A007661(n)-1),print1(n","))) \\ _M. F. Hasler_, Nov 26 2007
%Y A084438 Cf. A007661, A135726, A037082, A037083.
%K A084438 hard,nonn
%O A084438 1,1
%A A084438 _Hugo Pfoertner_, Jun 25 2003
%E A084438 Missing 26 inserted by _M. F. Hasler_, Nov 26 2007
%E A084438 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008
%E A084438 Edited by _N. J. A. Sloane_, Feb 11 2009 at the suggestion of _M. F. Hasler_