A381460 Smallest n-th perfect power greater than 1 satisfying A373387(a(n)) = n.

2, 49, 15625, 625, 7737809375, 735091890625, 1253790880222890625, 6634204312890625, 47312447868976594992787109375, 72624607478879073313928212890625, 471781339858152691906169456697218212890625, 1344888824246298437178134918212890625

Offset

1,1

Comments

This sequence has infinitely many terms since a (trivial) upper bound for a(n) is given by (10^(n - v_{10}(n)) + 1)^n, where v_{10} corresponds to the number of trailing 0's of n (if any), and each of these terms is mandatorily a perfect power of degree n or a multiple of n (e.g., a(3) = 25^3 = 5^6).

All the a(n) are generated by the 13 nontrivial 10-adic solutions of the fundamental equation y^5 = y: ...480163574218751 (see A063006), ...263499879186432 (see A120817), ...996418333704193 (see A290375), ...476581907922943 (see A290373), ...259918212890624 (see A091664), ...259918212890625 (see A018247), ...740081787109375 (see A091663), ...740081787109376 (see A018248), ...003581666295807 (see A290372), ...523418092077057 (see A290374), ...736500120813568 (see A120818), ...519836425781249 (see A091661), and ...999999999999999.

The bases of a(11), a(12), ..., a(50) have been provided by Max Alekseyev on February 14, 2025 (see "Closed form for the general term of 2, 49, 15625, 625, ..." in Links).

It is conjectured that if n > 2 is given, then a(n) is generated by {5^2^k}_oo or -{5^2^k}_oo (this probabilistic argument is based on the study of a(n) up to n = 50 since a(50) is a 762 digit number generated by {5^2^k}_oo = ...6259918212890625).

Links

Table of n, a(n) for n=1..12.

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.

Marco Ripà, On the relation between perfect powers and tetration frozen digits, Journal of AppliedMath, 2024, 2(5), 1771.

Math Overflow, Closed form for the general term of 2, 49, 15625, 625, ....

Wikipedia, Tetration.

Example

a(3) = 15625 since 15625 = 25^3 and 15625 is the smallest perfect cube whose constant congruence speed equals 3.

Crossrefs

Cf. A018247, A018248, A063006, A091661, A091663, A091664, A120817, A120818, A290372, A290373, A290374, A290375, A317905, A373387, A379243.

Sequence in context: A269839 A088067 A145676 * A366672 A335406 A291188

Adjacent sequences: A381454 A381455 A381456 * A381461 A381464 A381465

Keyword

nonn,base,hard

Author

Marco Ripà, Feb 24 2025

Status

approved