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A034886 Number of digits in n!. 31
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."

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

Index entries for sequences related to factorial numbers.

Noam D. Elkies, A counterexample to Kamenetsky's formula for the number of digits in n-factorial.

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(2*pi*n)/2+n*(log(n/e))), where log is the logarithm base 10. This approximation gives the exact answer for 2<=n<=5*10^7. - Dmitry Kamenetsky, Jul 07 2008

a(n) = ceiling(log10(1) + log10(2) + ... + log10(n)), where log10 is the logarithm base 10. [From 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[10])] + 1; f[0] = f[1] = 1; Array[f, 72, 0] (* Robert G. Wilson v, Jan 09 2013 *)

PROG

(Haskell)

a034886 = a055642 . a000142  -- Reinhard Zumkeller, Apr 08 2012

CROSSREFS

Cf. A137580 (distinct digits).

Sequence in context: A101788 A024698 A011883 * A011882 A025767 A091848

Adjacent sequences:  A034883 A034884 A034885 * A034887 A034888 A034889

KEYWORD

nonn,base,easy

AUTHOR

Erich Friedman

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 September 20 16:22 EDT 2017. Contains 292276 sequences.