A046882 Ultrafactorials: a(n) = n!^n!.

1, 1, 4, 46656, 1333735776850284124449081472843776

Offset: 0

Author

Camillo Lamonaca (Camillo.Lamonaca(AT)dva.gov.au)

Keywords

• nonn

Comments

a(5) = 3175 042373 780336 892901 667920 556557 182493 442088 021222 004926 225128 381629 943118 937129 098831 435345 716937 405655 305190 657814 877412 786176 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000. - Jonathan Vos Post, Dec 09 2004

Note that, by analogy with factorial primes, subfactorial primes, superfactorial primes and hyperfactorial primes, if a(n)+1 or a(n)-1 is prime, it should be called an ultrafactorial prime. These begin: a(0)+1 = a(1)+1 = 2, a(2)-1 = 3, a(2)+1 = 5. Are there any more? Note that a(3) = 46657 = 13 * 37 * 97 is a 3-brilliant number. a(3)-5, a(3)-3 and a(3)+5 are semiprime; a(3)-7 and a(3)+7 are primes. - Jonathan Vos Post, Dec 09 2004

Links

• Jean-Christophe Pain, Bounds on the p-adic valuation of the factorial, hyperfactorial and superfactorial, arXiv:2408.00353 [math.NT], 2024. See p. 6.
• Eric Weisstein's World of Mathematics, Ultrafactorial.

Crossrefs

Cf. A002109, A100085.

Cf. A000166, A000178, A002982.

Programs

Mathematica

lst={};Do[a=n!^n!;AppendTo[lst, a], {n, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 01 2008 *)
#^#&/@(Range[0,5]!) (* Harvey P. Dale, Oct 15 2023 *)

Formula

Sum_{n>=1} 1/a(n) = A100085. - Amiram Eldar, Nov 11 2020