A336220 Perfect powers which are totients of factorials.

1, 8, 32, 9216, 82944, 8294400, 1194393600

Offset

1,2

Comments

Corresponding values of factorials are 1! (and 2!), 4!, 5!, 8!, 9!, 11! and 13!, respectively.

This sequence is complete by Saunders, Theorem 2.

More generally, Saunders, Theorem 2 states that, for any positive integers a, b, c, m with gcd(b, c) = 1, there are only finitely many solutions to phi(a*n!/b) = cx^m and these solutions satisfy n <= max {61, 3a, 3b, 3c}.

Links

Table of n, a(n) for n=1..7.

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(4) = 9216 = 96^2 and phi(8!) = phi(40320) = 9216.

Crossrefs

Cf. A000010 (totient), A000142 (factorial numbers), A001597 (perfect powers).

Sequence in context: A288457 A166995 A079271 * A247533 A240547 A031445

Adjacent sequences: A336217 A336218 A336219 * A336221 A336222 A336223

Keyword

nonn,fini,full

Author

Tomohiro Yamada, Jul 17 2020

Status

approved