This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A169661 #15 Jul 22 2025 07:30:54 %S A169661 1,2,6,720,5040,3628800,39916800 %N A169661 Compact factorials of positive integers. %C A169661 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 A169661 T. M. Apostol, <a href="https://zbmath.org/?q=an:1112.11006">Review of "Compact integers and factorials" by V. Shevelev</a>, zbMATH. %H A169661 Vladimir Shevelev, <a href="https://eudml.org/doc/277854">Compact integers and factorials</a>, Acta Arith. 126 (2007), no.3, 195-236. %F A169661 a(n) = A263881(n)!. - _Jonathan Sondow_, Nov 17 2015 %Y A169661 Cf. A050376, A169655, A263881. %K A169661 nonn,fini,full %O A169661 1,2 %A A169661 _Vladimir Shevelev_, Apr 05 2010, Jun 29 2010