cp's OEIS Frontend

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.