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

%I A381949 #10 Mar 18 2025 17:36:14
%S A381949 2,7,55,5,95,95,385,95,1535,1535,6145,1025,24575,24575,98305,4095,
%T A381949 393215,393215,1572865,262145,6291455,6291455,25165825,6291455,
%U A381949 100663295,100663295,402653185,67108865,1610612735,1610612735,6442450945,402653185,25769803775,25769803775
%N A381949 a(n) is the smallest integer k greater than 1 and not a perfect power satisfying A373387(k^n) = n.
%C A381949 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).
%C A381949 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).
%C A381949 It is conjectured that a(n) is a multiple of 5 for any n > 2 (probabilistic argument).
%H A381949 Math Overflow, <a href="https://mathoverflow.net/questions/487698/closed-form-for-the-general-term-of-2-49-15625-625-dotsc">Closed form for the general term of 2, 49, 15625, 625, ...</a>.
%H A381949 Marco Ripà, <a href="https://doi.org/10.7546/nntdm.2021.27.4.43-61">The congruence speed formula</a>, Notes on Number Theory and Discrete Mathematics, 2021, 27(4), 43—61.
%H A381949 Marco Ripà, <a href="https://www.researchgate.net publication/387314761_Twelve_Python_Programs_to_Help_Readers_Test_Peculiar_Properties_of_Integer_Tetration">Twelve Python Programs to Help Readers Test Peculiar Properties of Integer Tetration</a>, ResearchGate, 2024.
%H A381949 Marco Ripà and Luca Onnis, <a href="https://doi.org/10.7546/nntdm.2022.28.3.441-457">Number of stable digits of any integer tetration</a>, Notes on Number Theory and Discrete Mathematics, 2022, 28(3), 441—457.
%H A381949 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%e A381949 a(3) = 55 since 5*11 is not a perfect power and A373387(55^3) = 3.
%Y A381949 Cf. A018247, A091663, A317905, A373387, A381460.
%K A381949 base,hard,nonn
%O A381949 1,1
%A A381949 _Marco Ripà_, Mar 10 2025