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

1, 7, 158, 2568, 35660, 456574, 5565709, 65657060, 756570557, 8565705523, 95657055187, 1056570551816, 11565705518104, 125657055180975, 1356570551809683, 14565705518096757, 155657055180967491

Offset

0,2

References

Jerry Glynn and Theodore Gray, "The Beginner's Guide To Mathematica, Version 4," Cambridge University Press, Cambridge, UK, 2000, p. 26.

Links

Robert G. Wilson v, Table of n, a(n) for n = 0..1000 (first 201 terms from Enrique Pérez Herrero)

Enrique Pérez Herrero, Trailing Zeros in n!, Psychedelic Geometry Blogspot

Eric Weisstein's World of Mathematics, Factorial

Formula

From Enrique Pérez Herrero, Nov 09 2009: (Start)

a(n) = 1 + floor(log((10^n)!)/(log(10))), and using Stirling's approximation:

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)

a(n) = 10^n*(n - 1/log(10)) + n/2 + O(1). [Arkadiusz Wesolowski, Jan 21 2012]

Mathematica

Table[ Floor[ N[ Log[ 10, (10^n)! ]] + 1 ], {n, 0, 7} ]
$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 *)

Crossrefs

Cf. A000142, A034886, A008904.

Sequence in context: A009703 A014385 A114485 * A153714 A153713 A137995

Adjacent sequences: A061007 A061008 A061009 * A061011 A061012 A061013

Keyword

nonn,base

Author

Robert G. Wilson v, May 20 2001

Extensions

a(7) from Farideh Firoozbakht, Jul 05 2005

More terms from Eric W. Weisstein, Dec 01 2005

Typo in formula fixed, and Mathematica formula changed to cover a(0)=1, Enrique Pérez Herrero, Feb 06 2010

Status

approved