A016634 -id:A016634 - cp's OEIS frontend

A068461 Factorial, or factoradic, expansion of log(11) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.

2, 0, 2, 1, 2, 4, 3, 3, 1, 2, 4, 0, 3, 13, 1, 12, 12, 13, 1, 16, 16, 0, 16, 12, 10, 9, 1, 23, 3, 22, 0, 28, 11, 14, 23, 16, 0, 14, 6, 1, 1, 14, 4, 25, 43, 0, 29, 10, 41, 19, 47, 14, 0, 51, 10, 47, 37, 45, 46, 56, 57, 45, 10, 32, 61, 15, 9, 67, 5, 9, 22, 25, 65, 56, 24, 12, 71, 9, 57

Offset: 1

Benoit Cloitre, Mar 10 2002

nonn

			log(11) = 2 + 0/2! + 2/3! + 1/4! + 2/5! + 4/6! + 3/7! + 3/8! + 1/9! + ...

Cf. A016634 (decimal expansion), A016739 (continued fraction).
Cf. A007514 vs. A075874 for factoradic expansion.
Cf. A067882 (log(2)), A322334 (log(3)), A322333 (log(5)), A068460 (log(7)).

SetDefaultRealField(RealField(250));  [Log(11)] cat [Floor(Factorial(n)*Log(11)) - n*Floor(Factorial((n-1))*Log(11)) : n in [2..80]]; // G. C. Greubel, Dec 05 2018

With[{b = Log[11]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Dec 05 2018 *)

vector(30, n, if(n>1, t=t%1*n, t=log(11))\1) \\ Increase realprecision (e.g., \p500) to compute more terms. - M. F. Hasler, Nov 25 2018

default(realprecision, 250); b = log(11); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Dec 05 2018

def a(n):
    if n==1: return floor(log(11))
    else: return expand(floor(factorial(n)*log(11)) - n*floor(factorial(n-1)*log(11)))
[a(n) for n in (1..80)] # G. C. Greubel, Dec 05 2018

Name edited and keyword cons,easy removed by M. F. Hasler, Nov 26 2018

A016739 Continued fraction for log(11).

Original entry on oeis.org

2, 2, 1, 1, 18, 2, 2, 7, 1, 6, 1, 1, 4, 1, 32, 4, 1, 1, 1, 3, 1, 1, 1, 2, 1, 4, 1, 1, 3, 1, 43, 4, 1, 1, 1, 25, 3, 2, 17, 4, 2, 1, 2, 1, 3, 1, 1, 89, 5, 1, 1, 1, 3, 1, 1, 2, 15, 3, 1, 49, 1, 1, 2, 35, 23, 1, 7, 2, 2, 3, 1, 2, 2, 1, 74, 1, 2, 3, 2, 1, 6, 1, 36, 4

Offset: 0

N. J. A. Sloane

			2.39789527279837054406194357... = 2 + 1/(2 + 1/(1 + 1/(1 + 1/(18 + ...)))). - _Harry J. Smith_, May 16 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
G. Xiao, Contfrac

Cf. A016634 (decimal expansion).

ContinuedFraction(Log(11)); // G. C. Greubel, Sep 15 2018

ContinuedFraction[Log[11], 100] (* G. C. Greubel, Sep 15 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(11)); for (n=1, 20000, write("b016739.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 16 2009

Offset changed by Andrew Howroyd, Jul 10 2024

A016645 Decimal expansion of log(22).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.091042453358315853479175699423305867897206988297672429339247718623967....

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016450 (continued fraction).

RealDigits[Log[22],10,120][[1]] (* Harvey P. Dale, Aug 18 2014 *)

default(realprecision, 20080); x=log(22); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016645.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009

Equals A016634 + A002162. - R. J. Mathar, Jul 22 2025

A016667 Decimal expansion of log(44).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.784189633918261162896407820881482435972707122657927683459927728117360....

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016472 (continued fraction).

RealDigits[Log[44],10,120][[1]] (* Harvey P. Dale, Nov 06 2013 *)

default(realprecision, 20080); x=log(44); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016667.txt", n, " ", d)); \\ Harry J. Smith, May 21 2009

Equals 2*A002162 + A016634. - R. J. Mathar, Jun 07 2024

A016678 Decimal expansion of log(55).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			4.007333185232470918662702911191316939347308208205934897131215600604752....

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016483 (continued fraction).

RealDigits[Log[55], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(55); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016678.txt", n, " ", d)); \\ Harry J. Smith, May 22 2009

Equals A016628+A016634. - R. J. Mathar, Jun 07 2024

cp's OEIS Frontend

A068461 Factorial, or factoradic, expansion of log(11) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.

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A016739 Continued fraction for log(11).

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A016645 Decimal expansion of log(22).

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A016667 Decimal expansion of log(44).

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A016678 Decimal expansion of log(55).

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