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A353246 Integer part of e[n]e, where [n] indicates hyper-n and e = 2.718281828... (using H. Kneser's proposal for n > 3).

%I A353246 #11 Aug 11 2025 07:39:21
%S A353246 4,5,7,15,2075
%N A353246 Integer part of e[n]e, where [n] indicates hyper-n and e = 2.718281828... (using H. Kneser's proposal for n > 3).
%C A353246 The common hyperoperation sequence is defined as follows: hyper-0 = zeration, hyper-1 = addition, hyper-2 = multiplication, hyper-3 = exponentiation, hyper-4 = tetration, and so on...
%C A353246 Thus e[0]e = e + 2, e[1]e = 2*e, e[2]e = e^2, e[3]e = e^e, and so on.
%C A353246 The fifth term of the twin sequence of the present one, floor(Pi[4]Pi), is much larger than 2075 and it is harder to calculate, while the integer part of e[4]Pi is 37149801960 (17.9 million times bigger than a(4)).
%H A353246 Hellmuth Kneser, <a href="https://gdz.sub.uni-goettingen.de/id/PPN243919689_0187">Reelle analytische Lösungen der Gleichung phi(phi(x)) = e^x und verwandter Funktionalgleichungen</a>, J. reine angew. Math. 187, 56-67 (1950).
%H A353246 Sheldon Levenstein (user sheldonison), <a href="https://math.eretrandre.org/tetrationforum/showthread.php?tid=1017">New fatou.gp program</a>, Jul 10 2015, updated Aug 14 2019.
%H A353246 William Paulsen, <a href="http://myweb.astate.edu/wpaulsen/tetration.html">Tetration</a>.
%H A353246 William Paulsen, <a href="https://doi.org/10.1007/s10444-018-9615-7">Tetration for complex bases</a>, Advances in Computational Mathematics, Vol. 45, No. 1 (2019), pp. 243-267; <a href="https://www.researchgate.net/profile/William-Paulsen-2/publication/325532999_Tetration_for_complex_bases/links/5d88c9d992851ceb79346b5f/">ResearchGate link</a>.
%H A353246 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyperoperation#Definition,_most_common">Hyperoperation</a>
%F A353246 a(n) = floor(e[n]e).
%e A353246 For n = 3, a(3) = floor(e[3]e) = floor(e^e) = 15.
%Y A353246 Cf. A001113, A019762, A072334, A073226, A351727, A352396.
%K A353246 nonn,more,hard
%O A353246 0,1
%A A353246 _Marco Ripà_, Apr 08 2022