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%I #16 May 07 2022 20:08:24
%S 0,1,2,7,33,329
%N Smallest k such that k-th derivative of f_n(x) at x=1 is nonpositive, where f_{-1}(x)=0 and f_n(x)=x^(f_{n-1}(x)).
%C f_{-1}(x)=0,
%C f_0(x)=1,
%C f_1(x)=x,
%C f_n(x)=x^(x^(x^...)) is a power tower with n levels.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PowerTower.html">Power Tower</a>.
%t Table[Length[NestWhileList[Expand[D[#, x]] &, Nest[x^# &, 0, n + 1], (# /. x -> 1) > 0 &]] - 1, {n, -1, 3}]
%Y Cf. A005727, A179230, A179405, A179505, A277537.
%K nonn,hard,more
%O -1,3
%A _Vladimir Reshetnikov_, May 06 2022