A381949 a(n) is the smallest integer k greater than 1 and not a perfect power satisfying A373387(k^n) = n.

2, 7, 55, 5, 95, 95, 385, 95, 1535, 1535, 6145, 1025, 24575, 24575, 98305, 4095, 393215, 393215, 1572865, 262145, 6291455, 6291455, 25165825, 6291455, 100663295, 100663295, 402653185, 67108865, 1610612735, 1610612735, 6442450945, 402653185, 25769803775, 25769803775

Offset: 1

Author

Marco Ripà, Mar 10 2025

Keywords

• base
• hard
• nonn

Comments

The terms from a(11) to a(50) of this sequence have been provided by Max Alekseyev on February 14, 2025 (see "Closed form for the general term of 2, 49, 15625, 625, ..." in Links).

For n = 1, 2, ..., 50, A381460(n) equals a(n)^n with the sole exception of n = 3 (i.e., A381460(3) = (5*5)^3 <> 55^3 = a(3)^3, and this is the only known value of n such that A381460(n) <> a(n)^n).

It is conjectured that a(n) is a multiple of 5 for any n > 2 (probabilistic argument).

Examples

			a(3) = 55 since 5*11 is not a perfect power and A373387(55^3) = 3.
		

Links

• Math Overflow, Closed form for the general term of 2, 49, 15625, 625, ....
• Marco Ripà, The congruence speed formula, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
• Marco Ripà, Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration, ResearchGate, 2024.
• Marco Ripà and Luca Onnis, Number of stable digits of any integer tetration, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
• Wikipedia, Tetration.

Crossrefs

Cf. A018247, A091663, A317905, A373387, A381460.