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Compact factorials of positive integers
7

%I #10 Nov 17 2015 16:14:21

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

%N Compact factorials of positive integers

%C 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.

%H T. M. Apostol, <a href="https://zbmath.org/?q=an:1112.11006">Review of "Compact integers and factorials" by V. Shevelev</a>, zbMATH.

%H V. Shevelev, <a href="http://dx.doi.org/10.4064/aa126-3-1">Compact integers and factorials</a>, Acta Arith. 126 (2007), no.3, 195-236.

%F a(n) = A263881(n)!. - _Jonathan Sondow_, Nov 17 2015

%Y Cf. A050376, A169655, A263881.

%K nonn,fini,full

%O 1,2

%A _Vladimir Shevelev_, Apr 05 2010, Jun 29 2010