A213201 -id:A213201 - cp's OEIS frontend

A008905 Leading digit of n!.

1, 1, 2, 6, 2, 1, 7, 5, 4, 3, 3, 3, 4, 6, 8, 1, 2, 3, 6, 1, 2, 5, 1, 2, 6, 1, 4, 1, 3, 8, 2, 8, 2, 8, 2, 1, 3, 1, 5, 2, 8, 3, 1, 6, 2, 1, 5, 2, 1, 6, 3, 1, 8, 4, 2, 1, 7, 4, 2, 1, 8, 5, 3, 1, 1, 8, 5, 3, 2, 1, 1, 8, 6, 4, 3, 2, 1, 1, 1, 8, 7, 5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 9, 9, 9, 9, 1

Offset: 0

Russ Cox

Kunoff proved that the distribution of terms of this sequence follows Benford's law, i.e., the asymptotic density of terms with value d (between 1 and 9) is log_10(1+1/d). - Amiram Eldar, Sep 23 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

Cf. A000966, A000142, A018799, A202021 (leading digit of (10^n)!), A213201.

a008905 = a000030 . a000142  -- Reinhard Zumkeller, Apr 08 2012

f[n_] := Quotient[n!, 10^Floor@ Log[10, n!]]; Array[f, 105, 0]

a(n) = A000030(A000142(n)). - Reinhard Zumkeller, Apr 08 2012

Two less-efficient Mathematica codings removed by Robert G. Wilson v, Nov 05 2010

A008963 Initial digit of Fibonacci number F(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 1, 2, 3, 5, 8, 1, 2, 3, 6, 9, 1, 2, 4, 6, 1, 1, 2, 4, 7, 1, 1, 3, 5, 8, 1, 2, 3, 5, 9, 1, 2, 3, 6, 1, 1, 2, 4, 7, 1, 1, 2, 4, 7, 1, 2, 3, 5, 8, 1, 2, 3, 5, 9, 1, 2, 4, 6, 1, 1, 2, 4, 7, 1, 1, 3, 4, 8, 1, 2, 3, 5, 8, 1, 2, 3, 6, 9, 1, 2, 4, 6, 1, 1, 2, 4, 7, 1, 1, 3, 5, 8, 1

Offset: 0

N. J. A. Sloane

Benford's law applies since the Fibonacci sequence is of exponential growth: P(d)=log_10(1+1/d), in fact among first 5000 values the digit d=1 appears 1505 times, while 5000*P(1) is about 1505.15. - Carmine Suriano, Feb 14 2011

Wlodarski observed and Webb proved that the distribution of terms of this sequence follows Benford's law. - Amiram Eldar, Sep 23 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
William Webb, Distribution of the first digits of Fibonacci numbers, The Fibonacci Quarterly, Vol. 13, No. 4 (1975), pp. 334-336.
Wikipedia, Benford's law.
J. Wlodarski, Fibonacci and Lucas Numbers Tend to Obey Benford's Law, The Fibonacci Quarterly, Vol. 9, No. 1 (1971), pp. 87-88.
Index entries for sequences related to Benford's law.

Cf. A000045, A003893 (final digit).

Cf. A000030, A261607, A213201.

a008963 = a000030 . a000045  -- Reinhard Zumkeller, Sep 09 2015

F:= combinat[fibonacci]:
a:= n-> parse(""||(F(n))[1]):
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 22 2023

Table[IntegerDigits[Fibonacci[n]][[1]], {n, 0, 100}] (* T. D. Noe, Sep 23 2011 *)

vector(10001,n,f=fibonacci(n-1);f\10^(#Str(f)-1))

a(n) = A000030(A000045(n)). - Amiram Eldar, Sep 23 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{d=1..9} d*log(1+1/d)/log(10) = 3.440236... (A213201). - Amiram Eldar, Jan 12 2023
For n>5, a(n) = floor(10^{alpha*n-beta}), where alpha=log_10(phi), beta=log_10(5)/2, {x}=x-floor(x) denotes the fractional part of x, log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622. - Hans J. H. Tuenter, Aug 20 2025

A346640 Decimal expansion of 2 - log_3(2).

Original entry on oeis.org

1, 3, 6, 9, 0, 7, 0, 2, 4, 6, 4, 2, 8, 5, 4, 2, 5, 6, 2, 9, 0, 0, 4, 7, 2, 8, 8, 5, 6, 5, 7, 2, 3, 9, 1, 4, 5, 7, 0, 0, 4, 1, 4, 3, 5, 9, 8, 6, 8, 1, 1, 9, 5, 7, 2, 1, 2, 9, 3, 4, 5, 0, 5, 6, 1, 6, 1, 3, 1, 4, 7, 9, 8, 6, 1, 9, 0, 8, 5, 1, 9, 4, 9, 3, 8, 8, 2, 7, 3, 1, 1, 4, 5, 0, 5, 4, 8, 2, 5, 4, 4, 3, 8, 6, 4

Offset: 1

Kevin Ryde, Jul 26 2021

Bedford and McMullen show this is the metric dimension of Hironaka's curve and equivalent carpets (see A346639).

			1.3690702464285425629004728856572391...

Timothy Bedford, Crinkly Curves, Markov Partitions and Dimension, Ph.D. thesis, University of Warwick, 1984, see proposition 4.1, page 89, cap(E) for the case s=2, t=2, r=3, Sum(k_i)=3 (and noting log(t) is a multiplier, not an exponent).
Curtis T. McMullen, Hausdorff Dimension of General Sierpinski Carpets, Nagoya Mathematical Journal, volume 96, number 19, 1984, pages 1-9, see page 2, m.dim(R) for the case m = s = 2 and n = r = 3.

Cf. A102525, A346639.

Cf. A213201.

RealDigits[2 - Log[3, 2], 10, 105][[1]] (* Amiram Eldar, Jul 27 2021 *)

2 - log(2)/log(3) \\ Michel Marcus, Jul 27 2021

Equals 2 - A102525.

Equals Sum_{d=1..2} d*log(1+1/d)/log(3). Compare with A213201. - Michel Marcus, Dec 25 2022

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A008905 Leading digit of n!.

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A008963 Initial digit of Fibonacci number F(n).

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A346640 Decimal expansion of 2 - log_3(2).

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