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%I #13 Apr 19 2014 01:42:32
%S 1,6,3,6,6,2,6,2,0,7,7,8,0,9,2,3,7,7,0,6,6,3,9,2,3,4,8,9,7,2,1,8,3,5,
%T 0,2,1,8,2,4,4,1,7,1,6,0,2,9,9,4,1,7,0,8,6,8,5,8,7,4,2,6,0,0,5,8,9,0,
%U 2,0,9,6,4,6,0,3,9,5,8,5,9,7,3,6,5,1,9,7,1,8,1,0,6,0,0,8,7,6,2,0,3,9,1,5,0
%N Decimal expansion of the base x for which the double logarithm of 2 equals the natural log of 2, that is, log_x log_x 2 = log 2.
%e From _R. J. Mathar_, Oct 09 2010: (Start)
%e 1.63662620778092377066392348972183502182...
%e log_(1.63662..)(2) = 1.4070142427036...
%e log_(1.63662..)(1.407014..) = A002162. (End)
%p f := log(log(2))/log(x)-log(log(x))/log(x)-log(2) ; fz := x-f/diff(f,x) ; z := 1.6 ; Digits := 120 ; for i from 1 to 10 do z := evalf(subs(x=z,fz)) ; print(%) ; end do: # _R. J. Mathar_, Oct 09 2010
%t RealDigits[ Exp[ ProductLog[Log[2]^2] / Log[2]], 10, 105][[1]] (* _Jean-François Alcover_, Jan 28 2014 *)
%Y Cf. A030797, which is the decimal expansion of the base n for which the double logarithm of e (log_n log_n e) = log e = 1, and which is the inverse of LambertW(1).
%K nonn,cons
%O 1,2
%A _Geoffrey Caveney_, Oct 08 2010
%E More digits from _R. J. Mathar_, Oct 09 2010
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