A335723 Perfect powers whose totients are factorials.

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

Table of n, a(n) for n=1..9.

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 A167657 A175700

Adjacent sequences: A335720 A335721 A335722 * A335724 A335725 A335726

Keyword

nonn,fini,full

Author

Tomohiro Yamada, Jun 19 2020

Status

approved