%I #14 Aug 07 2020 16:55:19
%S 1,4,9,225,11025,176400,1587600,192099600,32464832400
%N Perfect powers whose totients are factorials.
%C Corresponding values of factorials are 1!, 2!, 3!, 5!, 7!, 8!, 9!, 11! and 13!, respectively.
%C This sequence is complete by Saunders, Theorem 1.
%C All integers in this sequence are square. Hence, there exists no m-th power with m >= 3 whose totient is a factorial except 1.
%C More generally, Saunders, Theorem 1 states that, for any positive integers a, b, c, m with gcd(b, c) = 1, there are only finitely many solutions to phi(ax^m) = b*n!/c and these solutions satisfy n <= max {61, 3a, 3b, 3c}.
%H J. C. Saunders, <a href="https://arxiv.org/abs/1902.01638">Diophantine equations involving the Euler totient function</a>, arXiv:1902.01638 [math.NT], 2019-2020.
%H J. C. Saunders, <a href="https://doi.org/10.1016/j.jnt.2019.09.001">Diophantine equations involving the Euler totient function</a>, J. Number Theory 209 (2020), 347-358.
%e a(5) = 11025 = 105^2 and phi(11025) = 5040 = 7!.
%Y Cf. A000010 (totient), A000142 (factorial numbers), A001597 (perfect powers).
%K nonn,fini,full
%O 1,2
%A _Tomohiro Yamada_, Jun 19 2020