This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A352396 #20 Aug 11 2025 05:28:13 %S A352396 4,5,8,23,37149801960 %N A352396 Integer part of e[n]Pi, where [n] indicates hyper-n, e = 2.718281828459045..., and Pi = 3.141592653589793... (using H. Kneser's proposal for n > 3). %C A352396 The first term of this sequence is given by floor(e[0]Pi) = floor(Pi + 1) = floor(4.14159) = 4, which is the integer part of "e zeration Pi". In general, zeration is not a commutative arithmetic operation, while floor(e[1]Pi) = floor(Pi + e) = floor(5.85987) = 5 and floor(e[2]Pi) = floor(Pi * e) = floor(8.53973) = 8 hold since e[1]Pi = Pi[1]e and e[2]Pi = Pi[2]e. %C A352396 If n = 3, then floor(e[3]Pi) = floor(e^Pi) = floor(23.14069) = 23 (if n > 2, then hyper-n is not characterized by the commutative property anymore, even if we can find fascinating examples as 4[3]2 = 2[3]4 = 16). %C A352396 Now, tetration can be extended to complex bases as described in the Paulsen reference and the corresponding term of the present sequence can be found using his online calculator (see Links), so we have that floor(e[4]Pi) = floor(37149801960.55) = 37149801960. An easy proof that 37149801960.55999 > e^^Pi > 37149801960.55 follows from the chain of inequalities 37149801960.5569855999 > |37149801960.5569855 + 5.9249049902894650649*10^(-11)| > e^^Pi > |37149801960.556985498 + 5.9249049902894650647*10^(-11)| > 37149801960.55. %C A352396 As far as we know, it has not been proved if e^^Pi is an irrational number (or not). %H A352396 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 A352396 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 A352396 William Paulsen, <a href="http://myweb.astate.edu/wpaulsen/tetration.html">Tetration</a>. %H A352396 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 A352396 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hyperoperation#Definition,_most_common">Hyperoperation</a> %H A352396 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a> %F A352396 a(n) = floor(e[n]Pi). %e A352396 For n = 3, a(3) = floor(e[3]Pi) = floor(e^Pi) = 15. %Y A352396 Cf. A000796, A001113, A019609, A039661, A059742, A351727, A353246. %K A352396 nonn,more,hard %O A352396 0,1 %A A352396 _Marco Ripà_, Apr 08 2022