A008905 -id:A008905 - cp's OEIS frontend

A136756 Leading digit of n! in base 5.

1, 1, 2, 1, 4, 4, 1, 1, 2, 4, 1, 4, 1, 1, 2, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 2, 3, 4, 1, 2, 3, 1, 2, 4, 1, 2, 1, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 2, 1, 4, 2, 1, 3, 2, 1, 3, 2, 1, 4, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 4, 2, 2, 1, 1, 4, 3, 2, 2, 1, 1

Offset: 0

Carl R. White, Jan 21 2008

			a(10) = 1 as 10! = 1412110200_5 which has leading digit 1. - _David A. Corneth_, Jan 15 2021

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A000142, A136692, A000012, A136754, A136755, A136757, A136758, A136759, A136760, A008905, A136761, A136762, A136763, A136764, A136765, A136766.

A136757 Leading digit of n! in base 6.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 3, 3, 5, 1, 2, 3, 1, 2, 1, 2, 1, 3, 1, 5, 3, 1, 1, 4, 2, 1, 1, 1, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 1, 1, 1, 2, 3, 4, 1, 1, 2, 3, 5, 1, 2, 3, 5, 1, 2, 4, 1, 2, 4, 1, 2, 4, 1, 2, 5, 1, 3, 1, 2, 5, 2, 4, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 2

Offset: 0

Carl R. White, Jan 21 2008

			a(10) = 2 as 10! = 205440000_6 which has leading digit 2. - _David A. Corneth_, Jan 15 2021

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A000142, A136693, A136754, A136755, A136756, A136758, A136759, A136760, A008905, A136761, A136762, A136763, A136764, A136765, A136766.

Table[First[IntegerDigits[n!,6]],{n,0,100}] (* Harvey P. Dale, Dec 13 2011 *)

A136764 a(n) = leading digit of n! in base 14.

Original entry on oeis.org

1, 1, 2, 6, 1, 8, 3, 1, 1, 9, 6, 5, 4, 4, 4, 4, 5, 6, 8, 10, 1, 1, 2, 4, 7, 13, 1, 3, 6, 1, 2, 4, 10, 1, 4, 11, 2, 5, 1, 2, 8, 1, 5, 1, 3, 11, 2, 9, 2, 7, 1, 7, 1, 7, 2, 7, 2, 9, 2, 11, 3, 1, 4, 1, 7, 2, 11, 3, 1, 6, 2, 11, 4, 1, 8, 3, 1, 6, 2, 1, 6, 2, 1, 6, 2, 1, 7, 3, 1, 9, 4, 2, 13, 6, 3, 1, 9, 4, 2, 1

Offset: 0

Carl R. White, Jan 21 2008

			a(10) = 6 as 10! = 6106640_14 which has leading digit 6. - _David A. Corneth_, Jan 15 2021

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A000142, A136700, A136754, A136755, A136756, A136757, A136758, A136759, A136760, A008905, A136761, A136762, A136763, A136765, A136766.

f[n_]:=First[IntegerDigits[n!,14]]; lst={};Do[AppendTo[lst,f[n]],{n,0,2*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)
Table[IntegerDigits[n!,14][[1]],{n,0,100}] (* Harvey P. Dale, Dec 24 2015 *)

a(n) = digits(n!, 14)[1]; \\ Michel Marcus, Jan 27 2015

A136765 a(n) = leading digit of n! in base 15.

Original entry on oeis.org

1, 1, 2, 6, 1, 8, 3, 1, 11, 7, 4, 3, 2, 2, 2, 2, 2, 2, 3, 4, 5, 7, 11, 1, 1, 3, 5, 9, 1, 2, 4, 9, 1, 3, 6, 1, 2, 6, 1, 2, 7, 1, 3, 10, 2, 6, 1, 4, 12, 2, 9, 2, 7, 1, 6, 1, 5, 1, 5, 1, 5, 1, 6, 1, 7, 2, 10, 2, 13, 4, 1, 6, 1, 9, 3, 1, 5, 1, 9, 3, 1, 6, 2, 12, 4, 1, 10, 3, 1, 9, 3, 1, 9, 3, 1, 10, 4, 1, 12, 5

Offset: 0

Carl R. White, Jan 21 2008

			a(10) = 4 as 10! = 41110300_15 which has leading digit 4. - _David A. Corneth_, Jan 15 2021

David A. Corneth, Table of n, a(n) for n = 0..9999

Cf. A000142, A136701, A136754, A136755, A136756, A136757, A136758, A136759, A136760, A008905, A136761, A136762, A136763, A136764, A136766.

f[n_]:=First[IntegerDigits[n!,15]]; lst={};Do[AppendTo[lst,f[n]],{n,0,2*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 14 2009 *)

a(n) = digits(n!, 15)[1]; \\ Michel Marcus, Jan 27 2015

A141053 Most-significant decimal digit of Fibonacci(5n+3).

Original entry on oeis.org

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

Offset: 0

Paul Curtz, Aug 01 2008

Leading digit of A134490(n).
From Johannes W. Meijer, Jul 06 2011: (Start)
The leading digit d, 1 <= d <= 9, of A141053 follows Benford’s Law. This law states that the probability for the leading digit is p(d) = log_10(1+1/d), see the examples.
We observe that the last digit of A134490(n), i.e. F(5*n+3) mod 10, leads to the Lucas sequence A000032(n) (mod 10), i.e. a repetitive sequence of 12 digits [2, 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9] with p(0) = p(5) = 0, p(1) = p(3) = p(7) = p(9) = 1/6 and p(2) = p(4) = p(6) = p(8) = 1/12. This does not obey Benford’s Law, which would predict that the last digit would satisfy p(d) = 1/10, see the links. (End)

			From _Johannes W. Meijer_, Jul 06 2011: (Start)
d     p(N=2000) p(N=4000) p(N=6000) p(Benford)
1      0.29900   0.29950   0.30033   0.30103
2      0.17700   0.17675   0.17650   0.17609
3      0.12550   0.12525   0.12517   0.12494
4      0.09650   0.09675   0.09700   0.09691
5      0.07950   0.07950   0.07933   0.07918
6      0.06700   0.06675   0.06700   0.06695
7      0.05800   0.05825   0.05800   0.05799
8      0.05150   0.05125   0.05100   0.05115
9      0.04600   0.04600   0.04567   0.04576
Total  1.00000   1.00000   1.00000   1.00000 (End)

Hans J. H. Tuenter, Table of n, a(n) for n = 0..10000
Kevin Brown, Benford's Law.
Eric Weisstein's World of Mathematics, Benford's Law.
Wikipedia, Benford's Law.
Index entries for sequences related to Benford's law

Cf. A000045 (F(n)), A008963 (Initial digit F(n)), A105511-A105519, A003893 (F(n) mod 10), A130893, A186190 (First digit tribonacci), A008952 (Leading digit 2^n), A008905 (Leading digit n!), A045510, A112420 (Leading digit Collatz 3*n+1 starting with 1117065), A007524 (log_10(2)), A104140 (1-log_10(9)). - Johannes W. Meijer, Jul 06 2011

Cf. A001622, A097348.

A134490 := proc(n) combinat[fibonacci](5*n+3) ; end proc:
A141053 := proc(n) convert(A134490(n),base,10) ; op(-1,%) ; end proc:
seq(A141053(n),n=0..70) ; # R. J. Mathar, Jul 04 2011

Table[IntegerDigits[Fibonacci[5n+3]][[1]],{n,0,70}] (* Harvey P. Dale, Jun 22 2025 *)

a(n) = floor(F(5*n+3)/10^(floor(log(F(5*n+3))/log(10)))). - Johannes W. Meijer, Jul 06 2011

For n>0, a(n) = floor(10^{alpha*n+beta}), where alpha=5*log_10(phi)-1, beta=log_10(1+2/sqrt(5)), {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 27 2025

Edited by Johannes W. Meijer, Jul 06 2011

A368010 Ordinal transform of the leading digit of the factorial numbers.

Original entry on oeis.org

1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 3, 2, 2, 1, 4, 3, 4, 3, 5, 4, 2, 6, 5, 4, 7, 3, 8, 5, 2, 6, 3, 7, 4, 8, 9, 6, 10, 3, 9, 5, 7, 11, 5, 10, 12, 4, 11, 13, 6, 8, 14, 6, 4, 12, 15, 2, 5, 13, 16, 7, 5, 9, 17, 18, 8, 6, 10, 14, 19, 20, 9, 7, 6, 11, 15, 21, 22, 23, 10

Offset: 0

Alois P. Heinz, Dec 07 2023

n is the a(n)-th nonnegative integer producing value A008905(n).

			a(11) = 3 because 11! = 39916800 is the third factorial with leading digit 3 after 9! = 362880 and 10! = 3628800.  A008905(k) = 3 for k = 9, 10, 11, ... .

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000142, A008905, A367799.

Ordinal transform of A008905.

a(n) = |{ j in {0..n} : A008905(j) = A008905(n) }|.

A202021 The leading digit of (10^n)!.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Jan 09 2013

I employed R. Wm. Gosper's approximation (A090583).

			(10^1)! = 3628800 begins with 3.
(10^6)! begins with 8 and (10^100)! begins with 1.

Google-tm Answers, mathtalk-ga on Apr 16 2005 19:20 PDT
Eric Weisstein's World of Mathematics, Stirling's Approximation.

Cf. A008905, A132826.

f[n_] := IntegerPart[ 10^FractionalPart[ N[(n*Log[n] - n + (1/2) Log[2 Pi*n + 1/3])/Log[10], 150]]]; f[1] = 1; Table[ f[10^n], {n, 0, 104}]

a(n)=my(g=lngamma(10^n+1)/log(10));g-=g\1;10^g\1 \\ Charles R Greathouse IV, Jan 09 2013

A332842 Initial digit of n-th superfactorial.

Original entry on oeis.org

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

Offset: 0

Eder Vanzei, Feb 26 2020

			a(6) = 2 because A000178(6) = 24883200.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A000030, A000178, A008905.

a(n) = digits(prod(k=2, n, k!))[1]; \\ Michel Marcus, Feb 26 2020

def A332842(n):
    m, k = 1, 1
    for i in range(2,n+1):
        k *= i
        m *= k
    return int(str(m)[0]) # Chai Wah Wu, Mar 17 2020

a(n) = A000030(A000178(n)).

A056113 Most significant digit of n-th primorial A002110.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Jul 28 2000

Cf. A008905.

Do[p = Product[Prime[m], {m, 1, n}]; Print[IntegerPart[p/10^Floor[N[Log[10, p], 12]]]], {n, 0, 121}]

cp's OEIS Frontend

A136756 Leading digit of n! in base 5.

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A136757 Leading digit of n! in base 6.

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A136764 a(n) = leading digit of n! in base 14.

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A136765 a(n) = leading digit of n! in base 15.

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A141053 Most-significant decimal digit of Fibonacci(5n+3).

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A368010 Ordinal transform of the leading digit of the factorial numbers.

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A202021 The leading digit of (10^n)!.

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A332842 Initial digit of n-th superfactorial.

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