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

Author

Robert G. Wilson v, May 20 2001

Keywords

• nonn
• base

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

Crossrefs

Cf. A000142, A034886, A008904.

Programs

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 *)

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]

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