login
A335723
Perfect powers whose totients are factorials.
0
1, 4, 9, 225, 11025, 176400, 1587600, 192099600, 32464832400
OFFSET
1,2
COMMENTS
Corresponding values of factorials are 1!, 2!, 3!, 5!, 7!, 8!, 9!, 11! and 13!, respectively.
This sequence is complete by Saunders, Theorem 1.
All integers in this sequence are square. Hence, there exists no m-th power with m >= 3 whose totient is a factorial except 1.
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}.
LINKS
J. C. Saunders, Diophantine equations involving the Euler totient function, arXiv:1902.01638 [math.NT], 2019-2020.
J. C. Saunders, Diophantine equations involving the Euler totient function, J. Number Theory 209 (2020), 347-358.
EXAMPLE
a(5) = 11025 = 105^2 and phi(11025) = 5040 = 7!.
CROSSREFS
Cf. A000010 (totient), A000142 (factorial numbers), A001597 (perfect powers).
Sequence in context: A286322 A318615 A030074 * A299122 A377665 A167657
KEYWORD
nonn,fini,full
AUTHOR
Tomohiro Yamada, Jun 19 2020
STATUS
approved