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

3, 4, 6, 8, 20, 26, 36, 50, 60, 114, 135, 138, 248, 315, 351, 429, 642, 5505, 8793, 12086, 13580, 23109, 34626, 34706, 56282, 57675, 58298

Offset: 1

Author

Hugo Pfoertner, Jun 25 2003

Keywords

• hard
• nonn

Comments

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).

Examples

			a(4) = 8 since 8!!! - 1 = 8*5*2 - 1 = 79 is the 4th prime of that form.
26!!! - 1 = 2504902399 is prime.
		

Links

• Ken Davis, Status of Search for Multifactorial Primes.
• Ken Davis, Results for n!3-1.
• Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
• ProthSearch.net (on web.archive.org), n3minus.txt.

Crossrefs

Cf. A007661, A135726, A037082, A037083.

Programs

Mathematica

multiFactorial[n_, k_] := If[n < 1, 1, n * multiFactorial[n - k, k]];
Select[Range[0, 1000], PrimeQ[multiFactorial[#, 3] - 1] & ] (* Robert Price, Apr 19 2019 *)
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 *)

PARI

A007661(n) = prod(i=1,(n-1)\3,n-=3,n+!n)
for(n=1,999,if(isprime(A007661(n)-1),print1(n","))) \\ M. F. Hasler, Nov 26 2007

Extensions

Missing 26 inserted by M. F. Hasler, Nov 26 2007

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 03 2008

Edited by N. J. A. Sloane, Feb 11 2009 at the suggestion of M. F. Hasler