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 A034886 Number of digits in n!. 37
 1, 1, 1, 1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 18, 19, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 88, 90, 91, 93, 95, 97, 99, 101, 102 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Most counterexamples to the Kamenetsky formula (see below) must belong to A177901. Noam D. Elkies reported on MathOverflow (see link): "A counterexample [to Kamenetsky's formula] is n_1 := 6561101970383, with log_10((n_1/e)^n_1*sqrt(2*Pi*n_1) = 81244041273652.999999999999995102483-, but log10(n_1!) = 81244041273653.000000000000000618508+. [...] n_1 is the first counterexample, and the only one up to 10^14." From Bernard Schott, Dec 07 2019: (Start) a(n) < n iff 2 <= n <= 21; a(n) = n iff n = 1, 22, 23, 24; a(n) > n iff n >= 25. (End) REFERENCES Gardner, M. "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 50-65, 1978 LINKS T. D. Noe, Table of n, a(n) for n = 0..10000 Wikipedia, Stirling's Formula. FORMULA a(n) = floor(log(n!)/log(10)) + 1. a(n) = A027869(n) + A079680(n) + A079714(n) + A079684(n) + A079688(n) + A079690(n) + A079691(n) + A079692(n) + A079693(n) + A079694(n); a(n) = A055642(A000142(n)). - Reinhard Zumkeller, Jan 27 2008 Using Stirling's formula we can derive an approximation, which is very fast to compute in practice: ceiling(log_10(2*Pi*n)/2 + n*(log_10(n/e))). This approximation gives the exact answer for 2 <= n <= 5*10^7. - Dmitry Kamenetsky, Jul 07 2008 a(n) = ceiling(log_10(1) + log_10(2) + ... + log_10(n)). - Dmitry Kamenetsky, Nov 05 2010 MAPLE A034886 := n -> `if`(n<2, 1, `if`(n<6561101970383, ceil((ln(2*Pi)-2*n+ln(n)*(1+2*n))/(2*ln(10))), length(n!))); # Peter Luschny, Aug 26 2011 MATHEMATICA Join[{1, 1}, Table[Ceiling[Log[10, 2 Pi n]/2 + n*Log[10, n/E]], {n, 2, 71}]] f[n_] := Floor[(Log[2Pi] - 2n + Log[n]*(1 + 2n))/(2Log)] + 1; f = f = 1; Array[f, 72, 0] (* Robert G. Wilson v, Jan 09 2013 *) PROG (Haskell) a034886 = a055642 . a000142  -- Reinhard Zumkeller, Apr 08 2012 (PARI) for(n=0, 30, print1(floor(log(n!)/log(10)) + 1, ", ")) \\ G. C. Greubel, Feb 26 2018 (MAGMA) [Floor(Log(Factorial(n))/Log(10)) + 1: n in [0..30]]; // G. C. Greubel, Feb 26 2018 CROSSREFS Cf. A006488 (a(n) is a square), A056851 (a(n) is a cube), A058814 (n divides a(n)). Cf. A137580 (number of distinct digits in n!), A027868 (number of trailing zeros in n!). Sequence in context: A101788 A024698 A011883 * A011882 A025767 A091848 Adjacent sequences:  A034883 A034884 A034885 * A034887 A034888 A034889 KEYWORD nonn,base,easy AUTHOR EXTENSIONS Explained that the formula is an approximation. Made the formula easier to read. - Dmitry Kamenetsky, Dec 15 2010 STATUS approved

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Last modified July 8 22:01 EDT 2020. Contains 335537 sequences. (Running on oeis4.)