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%I #7 Feb 17 2021 10:18:16
%S 6,9,0,3,4,7,1,2,6,1,1,4,9,6,4,3,1,9,4,6,7,3,2,8,4,3,8,4,6,4,1,8,9,4,
%T 2,4,4,3,9,8,3,3,1,9,7,3,8,2,7,2,6,7,0,0,2,8,9,6,1,3,1,9,1,6,4,3,6,5,
%U 0,1,5,3,5,2,8,9,1,1,5,3,3,4,9,3,8,6,7,7,1,3,2,9,5,5,0,2,8,4,4,5,8,2,4,7,9
%N Decimal expansion of the odd limit of the harmonic power tower (1/2)^(1/3)^...^(1/(2n+1)).
%C The harmonic power tower sequence is divergent in the sense that even and odd partial exponentials converge to distinct limits. [after Steven Finch]
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.11, p. 449.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>
%e 0.6903471261149643194673284384641894244398...
%t digits = 40; dn = 10; $RecursionLimit = 1000; Clear[h]; h[n_] := h[n] = Power @@ (1/Range[2, n]); h[dn + 1]; h[n = 2*dn + 1]; While[RealDigits[h[n], 10, digits] != RealDigits[h[n - dn], 10, digits], Print["n = ", n]; n = n + dn]; RealDigits[h[n], 10, digits] // First
%t digits = 120; difs = 1; sold = 0; n = 100; While[Abs[difs] > 10^(-digits - 5), s = N[1/(2*n + 1), 1000]; Do[s = 1/m^s, {m, 2*n, 2, -1}]; difs = s - sold; sold = s; n++]; RealDigits[s, 10, 120][[1]] (* _Vaclav Kotesovec_, Feb 17 2021 *)
%Y Cf. A242759.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, May 22 2014
%E More terms from _Alois P. Heinz_, May 22 2014
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