A098454 Limit of the power tower defined as follows: 2^(1/2); (2^(1/2))^(3^(1/3)); (2^(1/2))^((3^(1/3))^(4^(1/4))); etc.

1, 9, 4, 1, 4, 6, 1, 1, 2, 3, 5, 8, 2, 0, 7, 1, 6, 9, 1, 5, 1, 4, 9, 4, 8, 3, 7, 8, 1, 9, 8, 1, 2, 6, 2, 0, 4, 3, 6, 2, 9, 6, 8, 9, 2, 0, 6, 7, 8, 3, 1, 6, 6, 4, 6, 3, 0, 0, 8, 3, 9, 6, 5, 6, 2, 9, 9, 1, 4, 6, 9, 1, 9, 3, 1, 7, 4, 1, 9, 9, 1, 6, 2, 2, 8, 5, 0, 6, 0, 6, 3, 3, 0, 1, 7, 2, 5, 8, 5, 4, 0, 8, 4, 1, 8

Offset: 1

Author

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 08 2004

Keywords

• cons
• easy
• nonn

Examples

			1.941461123582071691514948378198126204362968920678316646300839656299146919...
		

Programs

Maple

a:=array(2..150): a[2]:=2^(1/2): for n from 3 to 150 do: m:=1: for p from n to 2 by -1 do: m:=(p^(1/p))^m: od: a[n]:=m: od: evalf(a[150],100);

Mathematica

f[n_] := Block[{k = n, e = 1}, While[k > 1, e = N[(k^(1/k))^e, 128]; k-- ]; e]; RealDigits[ f[105], 10, 105][[1]] (* Robert G. Wilson v, Sep 10 2004 *)

Formula

Let b(n)=n^(1/n). Let m=1, initially. For values of k from n to 2 in steps of -1, calculate m -> b(k)^m. This leads to the approximation of the constant starting at n^(1/n). The constant is the limit as n -> infinity.

Extensions

More terms from Robert G. Wilson v, Sep 10 2004

Offset corrected by R. J. Mathar, Feb 05 2009