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A061010 Number of digits in (10^n)!.

%I A061010 #46 Feb 16 2025 08:32:44
%S A061010 1,7,158,2568,35660,456574,5565709,65657060,756570557,8565705523,
%T A061010 95657055187,1056570551816,11565705518104,125657055180975,
%U A061010 1356570551809683,14565705518096757,155657055180967491
%N A061010 Number of digits in (10^n)!.
%D A061010 Jerry Glynn and Theodore Gray, "The Beginner's Guide To Mathematica, Version 4," Cambridge University Press, Cambridge, UK, 2000, p. 26.
%H A061010 Robert G. Wilson v, <a href="/A061010/b061010.txt">Table of n, a(n) for n = 0..1000</a> (first 201 terms from Enrique Pérez Herrero)
%H A061010 Enrique Pérez Herrero, <a href="http://psychedelic-geometry.blogspot.com/2009/08/trailing-zeros-in-n.html">Trailing Zeros in n!</a>, Psychedelic Geometry Blogspot
%H A061010 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Factorial.html">Factorial</a>
%F A061010 From _Enrique Pérez Herrero_, Nov 09 2009: (Start)
%F A061010 a(n) = 1 + floor(log((10^n)!)/(log(10))), and using Stirling's approximation:
%F A061010 a(n) = 1 + delta(n,0) + floor((-2*10^n + log(2) + (1+2*10^n)*n*log(10) + log(Pi))/(2*log(10))). (End)
%F A061010 a(n) = 10^n*(n - 1/log(10)) + n/2 + O(1). [_Arkadiusz Wesolowski_, Jan 21 2012]
%t A061010 Table[ Floor[ N[ Log[ 10, (10^n)! ]] + 1 ], {n, 0, 7} ]
%t A061010 $MaxPrecision = Infinity; A061010[n_] := 1 + KroneckerDelta[n, 0] + Floor[(-2*10^n + Log[2] + (1 + 2*10^n)*n*Log[10] + Log[Pi])/(2*Log[10])] (* _Enrique Pérez Herrero_, Nov 09 2009 *)
%Y A061010 Cf. A000142, A034886, A008904.
%K A061010 nonn,base
%O A061010 0,2
%A A061010 _Robert G. Wilson v_, May 20 2001
%E A061010 a(7) from _Farideh Firoozbakht_, Jul 05 2005
%E A061010 More terms from _Eric W. Weisstein_, Dec 01 2005
%E A061010 Typo in formula fixed, and Mathematica formula changed to cover a(0)=1, _Enrique Pérez Herrero_, Feb 06 2010