In http://mathoverflow.net/questions/19086, now closed, there was a mention of Dmitry Kamenetsky's formula, $[\bigl(\log(2\pi n)/2+n(\log n-\log e)\bigr)/\log 10]+1$, for the number of decimal digits in $n$-factorial. Here, $[x]$ is the integer part of $x$. The formula appears at A034886 in the Online Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/A034886. My question is whether this formula is exact for all $n$, or is it occasionally off. No proof of exactness is given at the OEIS, no paper of Kamenetsky appears in Math Reviews.

In the other thread, I mentioned the discussion in the Usenet newsgroup sci.math in January-February, Subject: Number of digits in factorial. Although neither proof nor counterexample was found, I'd recommend looking over that discussion before starting in on this question.

EDIT 11 Aug 2011: I note that the question also came up at m.se: question 8323, 30 Oct 2010.

In http://mathoverflow.net/questions/19086, now closed, there was a mention of Dmitry Kamenetsky's formula, $[\bigl(\log(2\pi n)/2+n(\log n-\log e)\bigr)/\log 10]+1$, for the number of decimal digits in $n$-factorial. Here, $[x]$ is the integer part of $x$. The formula appears at A034886 in the Online Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/A034886. My question is whether this formula is exact for all $n$, or is it occasionally off. No proof of exactness is given at the OEIS, no paper of Kamenetsky appears in Math Reviews.

In the other thread, I mentioned the discussion in the Usenet newsgroup sci.math in January-February, Subject: Number of digits in factorial. Although neither proof nor counterexample was found, I'd recommend looking over that discussion before starting in on this question.