A169661 Compact factorials of positive integers.

1, 2, 6, 720, 5040, 3628800, 39916800

Offset: 1

Author

Vladimir Shevelev, Apr 05 2010, Jun 29 2010

Keywords

• nonn
• fini
• full

Comments

A positive integer m is called a compact number if all factors of unique factorization of n over distinct terms of A050376 are relatively prime. It is convenient to suppose that 1 is compact number. Although the density of compact numbers is 0.872497..., it is easy to prove that the set of compact factorials is finite. Indeed, if n is sufficiently large, then the interval (n/4,n/3) contains a prime p and thus p^3||n! Therefore the factorization of n! over A050376 contains product p*p^2. Much more difficult to show that all compact factorials are: 1!,2!,3!,6!,7!,10!,11!. All these factorials are presented in the table.

Links

• T. M. Apostol, Review of "Compact integers and factorials" by V. Shevelev, zbMATH.
• Vladimir Shevelev, Compact integers and factorials, Acta Arith. 126 (2007), no.3, 195-236.

Crossrefs

Cf. A050376, A169655, A263881.

Formula

a(n) = A263881(n)!. - Jonathan Sondow, Nov 17 2015